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Theorem rabsn 4363
Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
Assertion
Ref Expression
rabsn (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2791 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 673 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 982 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43abbidv 2843 . 2 (𝐵𝐴 → {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)} = {𝑥𝑥 = 𝐵})
5 df-rab 3023 . 2 {𝑥𝐴𝑥 = 𝐵} = {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)}
6 df-sn 4286 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
74, 5, 63eqtr4g 2783 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  {cab 2710  {crab 3018  {csn 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-rab 3023  df-sn 4286
This theorem is referenced by:  unisn3  4561  sylow3lem6  18168  lineunray  32481  pmapat  35469  dia0  36760  nzss  38935  lco0  42643
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