Theorem c0exALT 39784

Description: Alternate proof of c0ex 10666 using more set theory axioms but fewer complex number axioms (add ax-10 2143, ax-11 2159, ax-13 2380, ax-nul 5177, and remove ax-1cn 10626, ax-icn 10627, ax-addcl 10628, and ax-mulcl 10630). (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 10636	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2768	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7185	1 ⊢ 0 ∈ V

Syntax hints:   ∈ wcel 2112  Vcvv 3410  (class class class)co 7151  0cc0 10568  1c1 10569  ici 10570   + caddc 10571   · cmul 10573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5177  ax-i2m1 10636

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-uni 4800  df-iota 6295  df-fv 6344  df-ov 7154

This theorem is referenced by: (None)