Theorem c0exALT 40210

Description: Alternate proof of c0ex 10900 using more set theory axioms but fewer complex number axioms (add ax-10 2139, ax-11 2156, ax-13 2372, ax-nul 5225, and remove ax-1cn 10860, ax-icn 10861, ax-addcl 10862, and ax-mulcl 10864). (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 10870	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2747	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7289	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3422  (class class class)co 7255  0cc0 10802  1c1 10803  ici 10804   + caddc 10805   · cmul 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225  ax-i2m1 10870

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by: (None)