Theorem s1prc 13766

Description: Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)

Assertion

Ref	Expression
s1prc	⊢ (¬ 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)

Proof of Theorem s1prc

Step	Hyp	Ref	Expression
1		ids1 13759	. 2 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
2		fvprc 6490	. . 3 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
3	2	s1eqd 13763	. 2 ⊢ (¬ 𝐴 ∈ V → ⟨“( I ‘𝐴)”⟩ = ⟨“∅”⟩)
4	1, 3	syl5eq 2821	1 ⊢ (¬ 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1508   ∈ wcel 2051  Vcvv 3410  ∅c0 4173   I cid 5308  ‘cfv 6186  ⟨“cs1 13757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-s1 13758

This theorem is referenced by: (None)