Theorem c0exALT 38048

Description: Alternate proof of c0ex 10357 using more set theory axioms but fewer complex number axioms (add ax-10 2192, ax-11 2207, ax-13 2389, ax-nul 5015, and remove ax-1cn 10317, ax-icn 10318, ax-addcl 10319, and ax-mulcl 10321). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
c0exALT	⊢ 0 ∈ V

Proof of Theorem c0exALT

Step	Hyp	Ref	Expression
1		ax-i2m1 10327	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2834	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 6943	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2164  Vcvv 3414  (class class class)co 6910  0cc0 10259  1c1 10260  ici 10261   + caddc 10262   · cmul 10264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015  ax-i2m1 10327

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-sn 4400  df-pr 4402  df-uni 4661  df-iota 6090  df-fv 6135  df-ov 6913

This theorem is referenced by: (None)