Theorem c0exALT 42233

Description: Alternate proof of c0ex 11247 using more set theory axioms but fewer complex number axioms (add ax-10 2137, ax-11 2153, ax-13 2373, ax-nul 5308, and remove ax-1cn 11205, ax-icn 11206, ax-addcl 11207, and ax-mulcl 11209). (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 11215	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2742	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7460	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2104  Vcvv 3477  (class class class)co 7426  0cc0 11147  1c1 11148  ici 11149   + caddc 11150   · cmul 11152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-nul 5308  ax-i2m1 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4916  df-iota 6511  df-fv 6567  df-ov 7429

This theorem is referenced by: (None)