Theorem c0exALT 43449

Description: Alternate proof of c0ex 11313 using more set theory axioms but fewer complex number axioms (add ax-10 2179, ax-11 2196, ax-13 2404, ax-nul 5263, and remove ax-1cn 11271, ax-icn 11272, ax-addcl 11273, and ax-mulcl 11275). (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 11281	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2772	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7464	1 ⊢ 0 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2146  Vcvv 3454  (class class class)co 7430  0cc0 11213  1c1 11214  ici 11215   + caddc 11216   · cmul 11218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263  ax-i2m1 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6501  df-fv 6555  df-ov 7433

This theorem is used by: (None)