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Theorem rabsnel 32376
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
Hypothesis
Ref Expression
rabsnel.1 𝐵 ∈ V
Assertion
Ref Expression
rabsnel ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnel
StepHypRef Expression
1 rabsnel.1 . . . 4 𝐵 ∈ V
21snid 4666 . . 3 𝐵 ∈ {𝐵}
3 eleq2 2814 . . 3 ({𝑥𝐴𝜑} = {𝐵} → (𝐵 ∈ {𝑥𝐴𝜑} ↔ 𝐵 ∈ {𝐵}))
42, 3mpbiri 257 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 elrabi 3673 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝐵𝐴)
64, 5syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  {csn 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-sn 4631
This theorem is referenced by:  ddemeas  33983
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