Theorem c0exALT 42251

Description: Alternate proof of c0ex 11237 using more set theory axioms but fewer complex number axioms (add ax-10 2140, ax-11 2156, ax-13 2375, ax-nul 5286, and remove ax-1cn 11195, ax-icn 11196, ax-addcl 11197, and ax-mulcl 11199). (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 11205	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2743	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7447	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3463  (class class class)co 7413  0cc0 11137  1c1 11138  ici 11139   + caddc 11140   · cmul 11142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5286  ax-i2m1 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-sn 4607  df-pr 4609  df-uni 4888  df-iota 6494  df-fv 6549  df-ov 7416

This theorem is referenced by: (None)