Theorem c0exALT 43048

Description: Alternate proof of c0ex 11204 using more set theory axioms but fewer complex number axioms (add ax-10 2176, ax-11 2192, ax-13 2404, ax-nul 5269, and remove ax-1cn 11162, ax-icn 11163, ax-addcl 11164, and ax-mulcl 11166). (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 11172	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2772	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7444	1 ⊢ 0 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2143  Vcvv 3455  (class class class)co 7410  0cc0 11104  1c1 11105  ici 11106   + caddc 11107   · cmul 11109

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-i2m1 11172

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4590  df-pr 4592  df-uni 4873  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is used by: (None)