Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cflem Unicode version

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
StepHypRef Expression
1 ssid 3522 . . 3
2 ssid 3522 . . . . 5
3 sseq2 3525 . . . . . 6
43rspcev 3210 . . . . 5
52, 4mpan2 671 . . . 4
65rgen 2817 . . 3
7 sseq1 3524 . . . . 5
8 rexeq 3055 . . . . . 6
98ralbidv 2896 . . . . 5
107, 9anbi12d 710 . . . 4
1110spcegv 3195 . . 3
121, 6, 11mp2ani 678 . 2
13 fvex 5881 . . . . . 6
1413isseti 3115 . . . . 5
15 19.41v 1771 . . . . 5
1614, 15mpbiran 918 . . . 4
1716exbii 1667 . . 3
18 excom 1849 . . 3
1917, 18bitr3i 251 . 2
2012, 19sylib 196 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator