Theorem c0exALT 42449

Description: Alternate proof of c0ex 11124 using more set theory axioms but fewer complex number axioms (add ax-10 2146, ax-11 2162, ax-13 2374, ax-nul 5249, and remove ax-1cn 11082, ax-icn 11083, ax-addcl 11084, and ax-mulcl 11086). (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 11092	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2743	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7390	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3438  (class class class)co 7356  0cc0 11024  1c1 11025  ici 11026   + caddc 11027   · cmul 11029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249  ax-i2m1 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-uni 4862  df-iota 6446  df-fv 6498  df-ov 7359

This theorem is referenced by: (None)