Theorem c0exALT 42293

Description: Alternate proof of c0ex 11255 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2157, ax-13 2377, ax-nul 5306, and remove ax-1cn 11213, ax-icn 11214, ax-addcl 11215, and ax-mulcl 11217). (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 11223	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2746	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7465	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3480  (class class class)co 7431  0cc0 11155  1c1 11156  ici 11157   + caddc 11158   · cmul 11160

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 2708  ax-nul 5306  ax-i2m1 11223

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by: (None)