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Theorem unimax 4949
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 4018 . . 3 𝐴𝐴
2 sseq1 4021 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
32elrab3 3696 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝑥𝐴} ↔ 𝐴𝐴))
41, 3mpbiri 258 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝐵𝑥𝐴})
5 sseq1 4021 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65elrab 3695 . . . 4 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
76simprbi 496 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
87rgen 3061 . 2 𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴
9 ssunieq 4948 . . 3 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → 𝐴 = {𝑥𝐵𝑥𝐴})
109eqcomd 2741 . 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 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-ss 3980  df-uni 4913
This theorem is referenced by:  lssuni  20955  chsupid  31441  shatomistici  32390  lssats  38994  lpssat  38995  lssatle  38997  lssat  38998  mrelatglbALT  48785  mreclat  48786  toplatmeet  48792
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