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Theorem ssrab2f 41368
Description: Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
ssrab2f.1 𝑥𝐴
Assertion
Ref Expression
ssrab2f {𝑥𝐴𝜑} ⊆ 𝐴

Proof of Theorem ssrab2f
StepHypRef Expression
1 nfrab1 3383 . . 3 𝑥{𝑥𝐴𝜑}
2 ssrab2f.1 . . 3 𝑥𝐴
31, 2dfss3f 3957 . 2 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥𝐴𝜑}𝑥𝐴)
4 rabidim1 3379 . 2 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mprgbir 3151 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wnfc 2959  {crab 3140  wss 3934
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-in 3941  df-ss 3950
This theorem is referenced by:  dmmptssf  41486  mptssid  41495  fnlimfvre  41939  limsupequzmpt2  41983  liminfequzmpt2  42056  smflimlem2  43033  smflim  43038  smfpimcclem  43066  smfsupxr  43075
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