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Theorem unimax 4944
Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
Assertion
Ref Expression
unimax (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unimax
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssid 4006 . . 3 𝐴𝐴
2 sseq1 4009 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
32elrab3 3693 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝑥𝐴} ↔ 𝐴𝐴))
41, 3mpbiri 258 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝐵𝑥𝐴})
5 sseq1 4009 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65elrab 3692 . . . 4 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
76simprbi 496 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
87rgen 3063 . 2 𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴
9 ssunieq 4943 . . 3 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → 𝐴 = {𝑥𝐵𝑥𝐴})
109eqcomd 2743 . 2 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → {𝑥𝐵𝑥𝐴} = 𝐴)
114, 8, 10sylancl 586 1 (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951   cuni 4907
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-ss 3968  df-uni 4908
This theorem is referenced by:  lssuni  20937  chsupid  31431  shatomistici  32380  lssats  39013  lpssat  39014  lssatle  39016  lssat  39017  mrelatglbALT  48885  mreclat  48886  toplatmeet  48892
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