Theorem c0exALT 42868

Description: Alternate proof of c0ex 11173 using more set theory axioms but fewer complex number axioms (add ax-10 2175, ax-11 2191, ax-13 2403, ax-nul 5256, and remove ax-1cn 11131, ax-icn 11132, ax-addcl 11133, and ax-mulcl 11135). (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 11141	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2771	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7430	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2142  Vcvv 3454  (class class class)co 7396  0cc0 11073  1c1 11074  ici 11075   + caddc 11076   · cmul 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256  ax-i2m1 11141

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by: (None)