Theorem c0exALT 43133

Description: Alternate proof of c0ex 11246 using more set theory axioms but fewer complex number axioms (add ax-10 2178, ax-11 2194, ax-13 2401, ax-nul 5263, and remove ax-1cn 11204, ax-icn 11205, ax-addcl 11206, and ax-mulcl 11208). (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 11214	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2769	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7449	1 ⊢ 0 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3450  (class class class)co 7415  0cc0 11146  1c1 11147  ici 11148   + caddc 11149   · cmul 11151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263  ax-i2m1 11214

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6490  df-fv 6542  df-ov 7418

This theorem is used by: (None)