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Theorem s1prc 14541
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
StepHypRef Expression
1 ids1 14534 . 2 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
2 fvprc 6873 . . 3 𝐴 ∈ V → ( I ‘𝐴) = ∅)
32s1eqd 14538 . 2 𝐴 ∈ V → ⟨“( I ‘𝐴)”⟩ = ⟨“∅”⟩)
41, 3eqtrid 2785 1 𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  c0 4320   I cid 5569  cfv 6535  ⟨“cs1 14532
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-s1 14533
This theorem is referenced by: (None)
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