Theorem oneqmini 4938

Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)

Assertion

Ref	Expression
oneqmini	|- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssint 4304	. . . . . 6 |- ( A C_ |^| B <-> A. x e. B A C_ x )
2		ssel 3493	. . . . . . . . . . . 12 |- ( B C_ On -> ( A e. B -> A e. On ) )
3		ssel 3493	. . . . . . . . . . . 12 |- ( B C_ On -> ( x e. B -> x e. On ) )
4	2, 3	anim12d 563	. . . . . . . . . . 11 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A e. On /\ x e. On ) ) )
5		ontri1 4921	. . . . . . . . . . 11 |- ( ( A e. On /\ x e. On ) -> ( A C_ x <-> -. x e. A ) )
6	4, 5	syl6 33	. . . . . . . . . 10 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A C_ x <-> -. x e. A ) ) )
7	6	expdimp 437	. . . . . . . . 9 |- ( ( B C_ On /\ A e. B ) -> ( x e. B -> ( A C_ x <-> -. x e. A ) ) )
8	7	pm5.74d 247	. . . . . . . 8 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. B -> -. x e. A ) ) )
9		con2b 334	. . . . . . . 8 |- ( ( x e. B -> -. x e. A ) <-> ( x e. A -> -. x e. B ) )
10	8, 9	syl6bb 261	. . . . . . 7 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. A -> -. x e. B ) ) )
11	10	ralbidv2 2892	. . . . . 6 |- ( ( B C_ On /\ A e. B ) -> ( A. x e. B A C_ x <-> A. x e. A -. x e. B ) )
12	1, 11	syl5bb 257	. . . . 5 |- ( ( B C_ On /\ A e. B ) -> ( A C_ |^| B <-> A. x e. A -. x e. B ) )
13	12	biimprd 223	. . . 4 |- ( ( B C_ On /\ A e. B ) -> ( A. x e. A -. x e. B -> A C_ |^| B ) )
14	13	expimpd 603	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A C_ |^| B ) )
15		intss1 4303	. . . . 5 |- ( A e. B -> |^| B C_ A )
16	15	a1i 11	. . . 4 |- ( B C_ On -> ( A e. B -> |^| B C_ A ) )
17	16	adantrd 468	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> |^| B C_ A ) )
18	14, 17	jcad 533	. 2 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> ( A C_ |^| B /\ |^| B C_ A ) ) )
19		eqss 3514	. 2 |- ( A = |^| B <-> ( A C_ |^| B /\ |^| B C_ A ) )
20	18, 19	syl6ibr 227	1 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   C_ wss 3471   |^|cint 4288   Oncon0 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891

This theorem is referenced by:  oneqmin  6639  alephval3  8508  cfsuc  8654  alephval2  8964