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Theorem unimax 3925
 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 B x A} = 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 3290 . . 3 A A
2 sseq1 3292 . . . 4 (x = A → (x AA A))
32elrab3 2995 . . 3 (A B → (A {x B x A} ↔ A A))
41, 3mpbiri 224 . 2 (A BA {x B x A})
5 sseq1 3292 . . . . 5 (x = y → (x Ay A))
65elrab 2994 . . . 4 (y {x B x A} ↔ (y B y A))
76simprbi 450 . . 3 (y {x B x A} → y A)
87rgen 2679 . 2 y {x B x A}y A
9 ssunieq 3924 . . 3 ((A {x B x A} y {x B x A}y A) → A = {x B x A})
109eqcomd 2358 . 2 ((A {x B x A} y {x B x A}y A) → {x B x A} = A)
114, 8, 10sylancl 643 1 (A B{x B x A} = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {crab 2618   ⊆ wss 3257  ∪cuni 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by: (None)
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