Theorem c0exALT 42910

Description: Alternate proof of c0ex 11200 using more set theory axioms but fewer complex number axioms (add ax-10 2182, ax-11 2198, ax-13 2410, ax-nul 5271, and remove ax-1cn 11158, ax-icn 11159, ax-addcl 11160, and ax-mulcl 11162). (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 11168	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2778	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7445	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2149  Vcvv 3463  (class class class)co 7411  0cc0 11100  1c1 11101  ici 11102   + caddc 11103   · cmul 11105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271  ax-i2m1 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-uni 4877  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by: (None)