Theorem intmin 2557

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 2083	. . . . . 6 |- A (_ A
2		sseq2 2086	. . . . . . . 8 |- (x = A -> (A (_ x <-> A (_ A))
3		eleq2 1538	. . . . . . . 8 |- (x = A -> (y e. x <-> y e. A))
4	2, 3	imbi12d 628	. . . . . . 7 |- (x = A -> ((A (_ x -> y e. x) <-> (A (_ A -> y e. A)))
5	4	rcla4v 1876	. . . . . 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 1816	. . . . . 6 |- y e. V
8	7	elintrab 2549	. . . . 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 2073	. . 3 |- (A e. B -> |^|{x e. B | A (_ x} (_ A)
11		ssintub 2555	. . 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 2080	. 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 958   e. wcel 960  A.wral 1648  {crab 1651   (_ wss 2050  |^|cint 2537

This theorem is referenced by:  intmin2 2561  bm2.5ii 3025  onsucmin 3078  rankonid 4705  rankr1id 4707  rankval4 4712  cldcls 7679  chsupid 9306  spanid 9312

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-v 1815  df-in 2054  df-ss 2056  df-int 2538

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