Theorem s1prc 14638

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 14631	. 2 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
2		fvprc 6871	. . 3 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
3	2	s1eqd 14635	. 2 ⊢ (¬ 𝐴 ∈ V → ⟨“( I ‘𝐴)”⟩ = ⟨“∅”⟩)
4	1, 3	eqtrid 2816	1 ⊢ (¬ 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294   I cid 5553  ‘cfv 6534  ⟨“cs1 14629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-s1 14630

This theorem is referenced by: (None)