Theorem c0exALT 40289

Description: Alternate proof of c0ex 10969 using more set theory axioms but fewer complex number axioms (add ax-10 2137, ax-11 2154, ax-13 2372, ax-nul 5230, and remove ax-1cn 10929, ax-icn 10930, ax-addcl 10931, and ax-mulcl 10933). (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 10939	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2747	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7309	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3432  (class class class)co 7275  0cc0 10871  1c1 10872  ici 10873   + caddc 10874   · cmul 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230  ax-i2m1 10939

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by: (None)