Theorem intmin 2553

Description: Any member of a class is the smallest of those members that include it.

Assertion

Ref	Expression
intmin	|- (A e. B -> |^|{x e. B | A (_ x} = A)

Distinct variable groups:   x,A   x,B

Proof of Theorem intmin

Step	Hyp	Ref	Expression
1		ssid 2080	. . . . . 6 |- A (_ A
2		sseq2 2083	. . . . . . . 8 |- (x = A -> (A (_ x <-> A (_ A))
3		eleq2 1535	. . . . . . . 8 |- (x = A -> (y e. x <-> y e. A))
4	2, 3	imbi12d 626	. . . . . . 7 |- (x = A -> ((A (_ x -> y e. x) <-> (A (_ A -> y e. A)))
5	4	rcla4v 1873	. . . . . 6 |- (A e. B -> (A.x e. B (A (_ x -> y e. x) -> (A (_ A -> y e. A)))
6	1, 5	mpii 45	. . . . 5 |- (A e. B -> (A.x e. B (A (_ x -> y e. x) -> y e. A))
7		visset 1813	. . . . . 6 |- y e. V
8	7	elintrab 2545	. . . . 5 |- (y e. |^|{x e. B | A (_ x} <-> A.x e. B (A (_ x -> y e. x))
9	6, 8	syl5ib 206	. . . 4 |- (A e. B -> (y e. |^|{x e. B | A (_ x} -> y e. A))
10	9	ssrdv 2070	. . 3 |- (A e. B -> |^|{x e. B | A (_ x} (_ A)
11		ssintub 2551	. . 3 |- A (_ |^|{x e. B | A (_ x}
12	10, 11	jctir 293	. 2 |- (A e. B -> (|^|{x e. B | A (_ x} (_ A /\ A (_ |^|{x e. B | A (_ x}))
13		eqss 2077	. 2 |- (|^|{x e. B | A (_ x} = A <-> (|^|{x e. B | A (_ x} (_ A /\ A (_ |^|{x e. B | A (_ x}))
14	12, 13	sylibr 200	1 |- (A e. B -> |^|{x e. B | A (_ x} = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047  |^|cint 2533

This theorem is referenced by:  intmin2 2557  bm2.5ii 3019  onsucmin 3072  rankonid 4695  rankr1id 4697  rankval4 4702  cldcls 7682  chsupid 9311  spanid 9317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-v 1812  df-in 2051  df-ss 2053  df-int 2534

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