Theorem c0exALT 42247

Description: Alternate proof of c0ex 11175 using more set theory axioms but fewer complex number axioms (add ax-10 2142, ax-11 2158, ax-13 2371, ax-nul 5264, and remove ax-1cn 11133, ax-icn 11134, ax-addcl 11135, and ax-mulcl 11137). (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 11143	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2739	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7424	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3450  (class class class)co 7390  0cc0 11075  1c1 11076  ici 11077   + caddc 11078   · cmul 11080

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264  ax-i2m1 11143

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 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-uni 4875  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by: (None)