Theorem oneqmini 5776

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 4493	. . . . . 6 |- ( A C_ |^| B <-> A. x e. B A C_ x )
2		ssel 3597	. . . . . . . . . . . 12 |- ( B C_ On -> ( A e. B -> A e. On ) )
3		ssel 3597	. . . . . . . . . . . 12 |- ( B C_ On -> ( x e. B -> x e. On ) )
4	2, 3	anim12d 586	. . . . . . . . . . 11 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A e. On /\ x e. On ) ) )
5		ontri1 5757	. . . . . . . . . . 11 |- ( ( A e. On /\ x e. On ) -> ( A C_ x <-> -. x e. A ) )
6	4, 5	syl6 35	. . . . . . . . . 10 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A C_ x <-> -. x e. A ) ) )
7	6	expdimp 453	. . . . . . . . 9 |- ( ( B C_ On /\ A e. B ) -> ( x e. B -> ( A C_ x <-> -. x e. A ) ) )
8	7	pm5.74d 262	. . . . . . . 8 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. B -> -. x e. A ) ) )
9		con2b 349	. . . . . . . 8 |- ( ( x e. B -> -. x e. A ) <-> ( x e. A -> -. x e. B ) )
10	8, 9	syl6bb 276	. . . . . . 7 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. A -> -. x e. B ) ) )
11	10	ralbidv2 2984	. . . . . 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 272	. . . . 5 |- ( ( B C_ On /\ A e. B ) -> ( A C_ |^| B <-> A. x e. A -. x e. B ) )
13	12	biimprd 238	. . . 4 |- ( ( B C_ On /\ A e. B ) -> ( A. x e. A -. x e. B -> A C_ |^| B ) )
14	13	expimpd 629	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A C_ |^| B ) )
15		intss1 4492	. . . . 5 |- ( A e. B -> |^| B C_ A )
16	15	a1i 11	. . . 4 |- ( B C_ On -> ( A e. B -> |^| B C_ A ) )
17	16	adantrd 484	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> |^| B C_ A ) )
18	14, 17	jcad 555	. 2 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> ( A C_ |^| B /\ |^| B C_ A ) ) )
19		eqss 3618	. 2 |- ( A = |^| B <-> ( A C_ |^| B /\ |^| B C_ A ) )
20	18, 19	syl6ibr 242	1 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |^|cint 4475   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  oneqmin  7005  alephval3  8933  cfsuc  9079  alephval2  9394