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Theorem unimax 4839
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 3940 . . 3 𝐴𝐴
2 sseq1 3943 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
32elrab3 3632 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝑥𝐴} ↔ 𝐴𝐴))
41, 3mpbiri 261 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝐵𝑥𝐴})
5 sseq1 3943 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65elrab 3631 . . . 4 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
76simprbi 500 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
87rgen 3119 . 2 𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴
9 ssunieq 4838 . . 3 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → 𝐴 = {𝑥𝐵𝑥𝐴})
109eqcomd 2807 . 2 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → {𝑥𝐵𝑥𝐴} = 𝐴)
114, 8, 10sylancl 589 1 (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  wss 3884   cuni 4803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804
This theorem is referenced by:  lssuni  19708  chsupid  29199  shatomistici  30148  lssats  36307  lpssat  36308  lssatle  36310  lssat  36311
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