Theorem s1prc 14576

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 14569	. 2 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
2		fvprc 6853	. . 3 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
3	2	s1eqd 14573	. 2 ⊢ (¬ 𝐴 ∈ V → ⟨“( I ‘𝐴)”⟩ = ⟨“∅”⟩)
4	1, 3	eqtrid 2777	1 ⊢ (¬ 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299   I cid 5535  ‘cfv 6514  ⟨“cs1 14567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-s1 14568

This theorem is referenced by: (None)