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Theorem unimax 3588
 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 (A B {x BxA} = 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 2941 . . 3 AA
2 sseq1 2943 . . . 4 (x = A → (xAAA))
32elrab3 2676 . . 3 (A B → (A {x BxA} ↔ AA))
41, 3mpbiri 157 . 2 (A BA {x BxA})
5 sseq1 2943 . . . . 5 (x = y → (xAyA))
65elrab 2675 . . . 4 (y {x BxA} ↔ (y B yA))
76simprbi 260 . . 3 (y {x BxA} → yA)
87rgen 2352 . 2 y {x BxA}yA
9 ssunieq 3587 . . 3 ((A {x BxA} y {x BxA}yA) → A = {x BxA})
109eqcomd 2027 . 2 ((A {x BxA} y {x BxA}yA) → {x BxA} = A)
114, 8, 10sylancl 394 1 (A B {x BxA} = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ∀wral 2284  {crab 2288   ⊆ wss 2894  ∪ cuni 3554 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555 This theorem is referenced by:  onuniss2  4187
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