Theorem c0exALT 42293

Description: Alternate proof of c0ex 11106 using more set theory axioms but fewer complex number axioms (add ax-10 2144, ax-11 2160, ax-13 2372, ax-nul 5242, and remove ax-1cn 11064, ax-icn 11065, ax-addcl 11066, and ax-mulcl 11068). (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 11074	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2740	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7380	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2111  Vcvv 3436  (class class class)co 7346  0cc0 11006  1c1 11007  ici 11008   + caddc 11009   · cmul 11011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242  ax-i2m1 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-uni 4857  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by: (None)