Theorem c0exALT 41476

Description: Alternate proof of c0ex 11213 using more set theory axioms but fewer complex number axioms (add ax-10 2136, ax-11 2153, ax-13 2370, ax-nul 5306, and remove ax-1cn 11171, ax-icn 11172, ax-addcl 11173, and ax-mulcl 11175). (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 11181	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2740	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7446	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473  (class class class)co 7412  0cc0 11113  1c1 11114  ici 11115   + caddc 11116   · cmul 11118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306  ax-i2m1 11181

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551  df-ov 7415

This theorem is referenced by: (None)