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Theorem unimax 3605
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 2958 . . 3 AA
2 sseq1 2960 . . . 4 (x = A → (xAAA))
32elrab3 2693 . . 3 (A B → (A {x BxA} ↔ AA))
41, 3mpbiri 157 . 2 (A BA {x BxA})
5 sseq1 2960 . . . . 5 (x = y → (xAyA))
65elrab 2692 . . . 4 (y {x BxA} ↔ (y B yA))
76simprbi 260 . . 3 (y {x BxA} → yA)
87rgen 2368 . 2 y {x BxA}yA
9 ssunieq 3604 . . 3 ((A {x BxA} y {x BxA}yA) → A = {x BxA})
109eqcomd 2042 . 2 ((A {x BxA} y {x BxA}yA) → {x BxA} = A)
114, 8, 10sylancl 392 1 (A B {x BxA} = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300  {crab 2304  wss 2911   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  onuniss2  4203
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