Theorem c0exALT 43083

Description: Alternate proof of c0ex 11220 using more set theory axioms but fewer complex number axioms (add ax-10 2179, ax-11 2195, ax-13 2406, ax-nul 5271, and remove ax-1cn 11178, ax-icn 11179, ax-addcl 11180, and ax-mulcl 11182). (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 11188	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2774	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7454	1 ⊢ 0 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2146  Vcvv 3457  (class class class)co 7420  0cc0 11120  1c1 11121  ici 11122   + caddc 11123   · cmul 11125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271  ax-i2m1 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-sn 4592  df-pr 4594  df-uni 4875  df-iota 6497  df-fv 6549  df-ov 7423

This theorem is used by: (None)