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Theorem sbth 7657
Description: Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set is smaller (has lower cardinality) than and vice-versa, then and are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7647 through sbthlem10 7656; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7656. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
sbth

Proof of Theorem sbth
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 7542 . . . 4
21brrelexi 5045 . . 3
31brrelexi 5045 . . 3
4 breq1 4455 . . . . . 6
5 breq2 4456 . . . . . 6
64, 5anbi12d 710 . . . . 5
7 breq1 4455 . . . . 5
86, 7imbi12d 320 . . . 4
9 breq2 4456 . . . . . 6
10 breq1 4455 . . . . . 6
119, 10anbi12d 710 . . . . 5
12 breq2 4456 . . . . 5
1311, 12imbi12d 320 . . . 4
14 vex 3112 . . . . 5
15 sseq1 3524 . . . . . . 7
16 imaeq2 5338 . . . . . . . . . 10
1716difeq2d 3621 . . . . . . . . 9
1817imaeq2d 5342 . . . . . . . 8
19 difeq2 3615 . . . . . . . 8
2018, 19sseq12d 3532 . . . . . . 7
2115, 20anbi12d 710 . . . . . 6
2221cbvabv 2600 . . . . 5
23 eqid 2457 . . . . 5
24 vex 3112 . . . . 5
2514, 22, 23, 24sbthlem10 7656 . . . 4
268, 13, 25vtocl2g 3171 . . 3
272, 3, 26syl2an 477 . 2
2827pm2.43i 47 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442   cvv 3109  \cdif 3472  u.cun 3473  C_wss 3475  U.cuni 4249   class class class wbr 4452  `'ccnv 5003  |`cres 5006  "cima 5007   cen 7533   cdom 7534
This theorem is referenced by:  sbthb  7658  sdomnsym  7662  domtriord  7683  xpen  7700  limenpsi  7712  php  7721  onomeneq  7727  unbnn  7796  infxpenlem  8412  fseqen  8429  infpwfien  8464  inffien  8465  alephdom  8483  mappwen  8514  infcdaabs  8607  infunabs  8608  infcda  8609  infdif  8610  infxpabs  8613  infmap2  8619  gchaleph  9070  gchhar  9078  inttsk  9173  inar1  9174  xpnnenOLD  13943  znnen  13946  qnnen  13947  rpnnen  13960  rexpen  13961  mreexfidimd  15047  acsinfdimd  15812  fislw  16645  opnreen  21336  ovolctb2  21903  vitali  22022  aannenlem3  22726  basellem4  23357  lgsqrlem4  23619  umgraex  24323  pellexlem4  30768  pellexlem5  30769  idomsubgmo  31155
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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-en 7537  df-dom 7538
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