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Theorem sbth 7776
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 7766 through sbthlem10 7775; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7775. 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 ((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 7658 . . . 4 Rel ≼
21brrelexi 4922 . . 3 (𝐴𝐵𝐴 ∈ V)
31brrelexi 4922 . . 3 (𝐵𝐴𝐵 ∈ V)
4 breq1 4437 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 4438 . . . . . 6 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 5anbi12d 734 . . . . 5 (𝑧 = 𝐴 → ((𝑧𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝐴)))
7 breq1 4437 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 329 . . . 4 (𝑧 = 𝐴 → (((𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 breq2 4438 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
10 breq1 4437 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
119, 10anbi12d 734 . . . . 5 (𝑤 = 𝐵 → ((𝐴𝑤𝑤𝐴) ↔ (𝐴𝐵𝐵𝐴)))
12 breq2 4438 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1311, 12imbi12d 329 . . . 4 (𝑤 = 𝐵 → (((𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
14 vex 3069 . . . . 5 𝑧 ∈ V
15 sseq1 3475 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
16 imaeq2 5214 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1716difeq2d 3576 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1817imaeq2d 5218 . . . . . . . 8 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
19 difeq2 3570 . . . . . . . 8 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2018, 19sseq12d 3483 . . . . . . 7 (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥)))
2115, 20anbi12d 734 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦)) ↔ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))))
2221cbvabv 2629 . . . . 5 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
23 eqid 2505 . . . . 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 3069 . . . . 5 𝑤 ∈ V
2514, 22, 23, 24sbthlem10 7775 . . . 4 ((𝑧𝑤𝑤𝑧) → 𝑧𝑤)
268, 13, 25vtocl2g 3132 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
272, 3, 26syl2an 487 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
2827pm2.43i 49 1 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 378   = wceq 1468  wcel 1937  {cab 2491  Vcvv 3066  cdif 3423  cun 3424  wss 3426   cuni 4228   class class class wbr 4434  ccnv 4879  cres 4882  cima 4883  cen 7649  cdom 7650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-en 7653  df-dom 7654
This theorem is referenced by:  sbthb  7777  sdomnsym  7781  domtriord  7802  xpen  7819  limenpsi  7831  php  7840  onomeneq  7846  unbnn  7912  infxpenlem  8529  fseqen  8543  infpwfien  8578  inffien  8579  alephdom  8597  mappwen  8628  infcdaabs  8721  infunabs  8722  infcda  8723  infdif  8724  infxpabs  8727  infmap2  8733  gchaleph  9181  gchhar  9189  inttsk  9284  inar1  9285  znnen  14425  qnnen  14426  rpnnen  14439  rexpen  14440  mreexfidimd  15722  acsinfdimd  16593  fislw  17438  opnreen  22007  ovolctb2  22604  vitali  22732  aannenlem3  23447  basellem4  24171  lgsqrlem4  24433  umgraex  25211  phpreu  32162  poimirlem26  32204  pellexlem4  35916  pellexlem5  35917  idomsubgmo  36312  upgrex  39718
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