Theorem c0exALT 42303

Description: Alternate proof of c0ex 11229 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2157, ax-13 2376, ax-nul 5276, and remove ax-1cn 11187, ax-icn 11188, ax-addcl 11189, and ax-mulcl 11191). (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 11197	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2744	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7439	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3459  (class class class)co 7405  0cc0 11129  1c1 11130  ici 11131   + caddc 11132   · cmul 11134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276  ax-i2m1 11197

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by: (None)