Theorem c0exALT 42691

Description: Alternate proof of c0ex 11138 using more set theory axioms but fewer complex number axioms (add ax-10 2147, ax-11 2163, ax-13 2376, ax-nul 5241, and remove ax-1cn 11096, ax-icn 11097, ax-addcl 11098, and ax-mulcl 11100). (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 11106	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2745	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7401	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3429  (class class class)co 7367  0cc0 11038  1c1 11039  ici 11040   + caddc 11041   · cmul 11043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241  ax-i2m1 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-sn 4568  df-pr 4570  df-uni 4851  df-iota 6454  df-fv 6506  df-ov 7370

This theorem is referenced by: (None)