Theorem oneqmini 4934

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

Distinct variable groups:   ,   ,

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssint 4302	. . . . . 6
2		ssel 3497	. . . . . . . . . . . 12
3		ssel 3497	. . . . . . . . . . . 12
4	2, 3	anim12d 563	. . . . . . . . . . 11
5		ontri1 4917	. . . . . . . . . . 11
6	4, 5	syl6 33	. . . . . . . . . 10
7	6	expdimp 437	. . . . . . . . 9
8	7	pm5.74d 247	. . . . . . . 8
9		con2b 334	. . . . . . . 8
10	8, 9	syl6bb 261	. . . . . . 7
11	10	ralbidv2 2892	. . . . . 6
12	1, 11	syl5bb 257	. . . . 5
13	12	biimprd 223	. . . 4
14	13	expimpd 603	. . 3
15		intss1 4301	. . . . 5
16	15	a1i 11	. . . 4
17	16	adantrd 468	. . 3
18	14, 17	jcad 533	. 2
19		eqss 3518	. 2
20	18, 19	syl6ibr 227	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  C_wss 3475  |^|cint 4286  con0 4883

This theorem is referenced by:  oneqmin  6640  alephval3  8512  cfsuc  8658  alephval2  8968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887