Theorem c0exALT 43303

Description: Alternate proof of c0ex 11300 using more set theory axioms but fewer complex number axioms (add ax-10 2178, ax-11 2194, ax-13 2402, ax-nul 5260, and remove ax-1cn 11258, ax-icn 11259, ax-addcl 11260, and ax-mulcl 11262). (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 11268	. . 3 ⊢ ((i · i) + 1) = 0
2	1	eqcomi 2770	. 2 ⊢ 0 = ((i · i) + 1)
3	2	ovexi 7454	1 ⊢ 0 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3451  (class class class)co 7420  0cc0 11200  1c1 11201  ici 11202   + caddc 11203   · cmul 11205

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 2733  ax-nul 5260  ax-i2m1 11268

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by: (None)