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But when I put the transformer, the light bulb doesn't lights.
I will make a suggestion. Try doing away with your 10 KHz oscillator and get your 555 making about a 50 / 60 Hz square wave. Drive the transformer 12 volt side with that low frequency square wave. Get rid of the caps and your light bulb because that isn't going to happen. Now place your neon bulb across the output (actually the 220 volt side of your transformer). Place a resistor in series with the neon bulb! Assume the bulb gas will ionize at around 90 to 100 volts, and assume you have 220 volts. Read a little on neon bulbs. Determine the resistor value. Now see if the bulb lights.
Ron