Theorem c0exALT 39460

Description: Alternate proof of c0ex 10624 using more set theory axioms but fewer complex number axioms (add ax-10 2142, ax-11 2158, ax-13 2379, ax-nul 5174, and remove ax-1cn 10584, ax-icn 10585, ax-addcl 10586, and ax-mulcl 10588). (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 10594	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2807	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7169	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2111  Vcvv 3441  (class class class)co 7135  0cc0 10526  1c1 10527  ici 10528   + caddc 10529   · cmul 10531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174  ax-i2m1 10594

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283  df-fv 6332  df-ov 7138

This theorem is referenced by: (None)