Theorem c0exALT 42736

Description: Alternate proof of c0ex 11129 using more set theory axioms but fewer complex number axioms (add ax-10 2152, ax-11 2168, ax-13 2380, ax-nul 5228, and remove ax-1cn 11087, ax-icn 11088, ax-addcl 11089, and ax-mulcl 11091). (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 11097	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2748	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7390	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2119  Vcvv 3431  (class class class)co 7356  0cc0 11029  1c1 11030  ici 11031   + caddc 11032   · cmul 11034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228  ax-i2m1 11097

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-sn 4556  df-pr 4558  df-uni 4839  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by: (None)