Theorem c0exALT 42228

Description: Alternate proof of c0ex 11128 using more set theory axioms but fewer complex number axioms (add ax-10 2142, ax-11 2158, ax-13 2370, ax-nul 5248, and remove ax-1cn 11086, ax-icn 11087, ax-addcl 11088, and ax-mulcl 11090). (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 11096	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2738	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7387	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3438  (class class class)co 7353  0cc0 11028  1c1 11029  ici 11030   + caddc 11031   · cmul 11033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248  ax-i2m1 11096

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4580  df-pr 4582  df-uni 4862  df-iota 6442  df-fv 6494  df-ov 7356

This theorem is referenced by: (None)