Theorem c0exALT 41170

Description: Alternate proof of c0ex 11204 using more set theory axioms but fewer complex number axioms (add ax-10 2137, ax-11 2154, ax-13 2371, ax-nul 5305, and remove ax-1cn 11164, ax-icn 11165, ax-addcl 11166, and ax-mulcl 11168). (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 11174	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2741	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7439	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3474  (class class class)co 7405  0cc0 11106  1c1 11107  ici 11108   + caddc 11109   · cmul 11111

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305  ax-i2m1 11174

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6492  df-fv 6548  df-ov 7408

This theorem is referenced by: (None)