Theorem cflem 8647

Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set . (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
cflem

Distinct variable group:   ,,,,

Proof of Theorem cflem

Step	Hyp	Ref	Expression
1		ssid 3522	. . 3
2		ssid 3522	. . . . 5
3		sseq2 3525	. . . . . 6
4	3	rspcev 3210	. . . . 5
5	2, 4	mpan2 671	. . . 4
6	5	rgen 2817	. . 3
7		sseq1 3524	. . . . 5
8		rexeq 3055	. . . . . 6
9	8	ralbidv 2896	. . . . 5
10	7, 9	anbi12d 710	. . . 4
11	10	spcegv 3195	. . 3
12	1, 6, 11	mp2ani 678	. 2
13		fvex 5881	. . . . . 6
14	13	isseti 3115	. . . . 5
15		19.41v 1771	. . . . 5
16	14, 15	mpbiran 918	. . . 4
17	16	exbii 1667	. . 3
18		excom 1849	. . 3
19	17, 18	bitr3i 251	. 2
20	12, 19	sylib 196	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  `cfv 5593  ccrd 8337

This theorem is referenced by:  cfval  8648  cff  8649  cff1  8659

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-nul 4581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-uni 4250  df-iota 5556  df-fv 5601