Theorem c0exALT 39229

Description: Alternate proof of c0ex 10628 using more set theory axioms but fewer complex number axioms (add ax-10 2144, ax-11 2160, ax-13 2389, ax-nul 5203, and remove ax-1cn 10588, ax-icn 10589, ax-addcl 10590, and ax-mulcl 10592). (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 10598	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2829	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7183	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3491  (class class class)co 7149  0cc0 10530  1c1 10531  ici 10532   + caddc 10533   · cmul 10535

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-nul 5203  ax-i2m1 10598

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-sn 4561  df-pr 4563  df-uni 4832  df-iota 6307  df-fv 6356  df-ov 7152

This theorem is referenced by: (None)