Theorem c0exALT 42239

Description: Alternate proof of c0ex 11278 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2158, ax-13 2380, ax-nul 5324, and remove ax-1cn 11236, ax-icn 11237, ax-addcl 11238, and ax-mulcl 11240). (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 11246	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2749	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7477	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488  (class class class)co 7443  0cc0 11178  1c1 11179  ici 11180   + caddc 11181   · cmul 11183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324  ax-i2m1 11246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6520  df-fv 6576  df-ov 7446

This theorem is referenced by: (None)