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Theorem intmin 3946
 Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
intmin (A B{x B A x} = A)
Distinct variable groups:   x,A   x,B

Proof of Theorem intmin
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . 5 y V
21elintrab 3938 . . . 4 (y {x B A x} ↔ x B (A xy x))
3 ssid 3290 . . . . 5 A A
4 sseq2 3293 . . . . . . 7 (x = A → (A xA A))
5 eleq2 2414 . . . . . . 7 (x = A → (y xy A))
64, 5imbi12d 311 . . . . . 6 (x = A → ((A xy x) ↔ (A Ay A)))
76rspcv 2951 . . . . 5 (A B → (x B (A xy x) → (A Ay A)))
83, 7mpii 39 . . . 4 (A B → (x B (A xy x) → y A))
92, 8syl5bi 208 . . 3 (A B → (y {x B A x} → y A))
109ssrdv 3278 . 2 (A B{x B A x} A)
11 ssintub 3944 . . 3 A {x B A x}
1211a1i 10 . 2 (A BA {x B A x})
1310, 12eqssd 3289 1 (A B{x B A x} = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {crab 2618   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-int 3927 This theorem is referenced by:  intmin2  3953
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