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Theorem sbth 8368
Description: Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 𝐴 is smaller (has lower cardinality) than 𝐵 and vice-versa, then 𝐴 and 𝐵 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 8358 through sbthlem10 8367; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 8367. 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. In the Intuitionistic Logic Explorer (ILE) the Schroeder-Bernstein Theorem has been proven equivalent to the law of the excluded middle (LEM), and in ILE the LEM is not accepted as necessarily true; see https://us.metamath.org/ileuni/exmidsbth.html. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
sbth ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)

Proof of Theorem sbth
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 8247 . . . 4 Rel ≼
21brrelex1i 5406 . . 3 (𝐴𝐵𝐴 ∈ V)
31brrelex1i 5406 . . 3 (𝐵𝐴𝐵 ∈ V)
4 breq1 4889 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 4890 . . . . . 6 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 5anbi12d 624 . . . . 5 (𝑧 = 𝐴 → ((𝑧𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝐴)))
7 breq1 4889 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 336 . . . 4 (𝑧 = 𝐴 → (((𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 breq2 4890 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
10 breq1 4889 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
119, 10anbi12d 624 . . . . 5 (𝑤 = 𝐵 → ((𝐴𝑤𝑤𝐴) ↔ (𝐴𝐵𝐵𝐴)))
12 breq2 4890 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1311, 12imbi12d 336 . . . 4 (𝑤 = 𝐵 → (((𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
14 vex 3400 . . . . 5 𝑧 ∈ V
15 sseq1 3844 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
16 imaeq2 5716 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1716difeq2d 3950 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1817imaeq2d 5720 . . . . . . . 8 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
19 difeq2 3944 . . . . . . . 8 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2018, 19sseq12d 3852 . . . . . . 7 (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥)))
2115, 20anbi12d 624 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦)) ↔ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))))
2221cbvabv 2913 . . . . 5 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
23 eqid 2777 . . . . 5 ((𝑓 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}) ∪ (𝑔 ↾ (𝑧 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}))) = ((𝑓 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}) ∪ (𝑔 ↾ (𝑧 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))})))
24 vex 3400 . . . . 5 𝑤 ∈ V
2514, 22, 23, 24sbthlem10 8367 . . . 4 ((𝑧𝑤𝑤𝑧) → 𝑧𝑤)
268, 13, 25vtocl2g 3470 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
272, 3, 26syl2an 589 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
2827pm2.43i 52 1 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  {cab 2762  Vcvv 3397  cdif 3788  cun 3789  wss 3791   cuni 4671   class class class wbr 4886  ccnv 5354  cres 5357  cima 5358  cen 8238  cdom 8239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-en 8242  df-dom 8243
This theorem is referenced by:  sbthb  8369  sdomnsym  8373  domtriord  8394  xpen  8411  limenpsi  8423  php  8432  onomeneq  8438  unbnn  8504  infxpenlem  9169  fseqen  9183  infpwfien  9218  inffien  9219  alephdom  9237  mappwen  9268  infcdaabs  9363  infunabs  9364  infcda  9365  infdif  9366  infxpabs  9369  infmap2  9375  gchaleph  9828  gchhar  9836  inttsk  9931  inar1  9932  znnen  15345  qnnen  15346  rpnnen  15360  rexpen  15361  mreexfidimd  16696  acsinfdimd  17568  fislw  18424  opnreen  23042  ovolctb2  23696  vitali  23817  aannenlem3  24522  basellem4  25262  lgsqrlem4  25526  upgrex  26440  phpreu  34013  poimirlem26  34056  pellexlem4  38349  pellexlem5  38350  idomsubgmo  38728
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