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Theorem sbth 8636
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. The theorem can also be proved from the axiom of choice and the linear order of the cardinal numbers, but our development does not provide the linear order of cardinal numbers until much later and in ways that depend on Schroeder-Bernstein.

The main proof consists of lemmas sbthlem1 8626 through sbthlem10 8635; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 8635. 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 8635. 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 8514 . . . 4 Rel ≼
21brrelex1i 5607 . . 3 (𝐴𝐵𝐴 ∈ V)
31brrelex1i 5607 . . 3 (𝐵𝐴𝐵 ∈ V)
4 breq1 5068 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 5069 . . . . . 6 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 5anbi12d 632 . . . . 5 (𝑧 = 𝐴 → ((𝑧𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝐴)))
7 breq1 5068 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 347 . . . 4 (𝑧 = 𝐴 → (((𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 breq2 5069 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
10 breq1 5068 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
119, 10anbi12d 632 . . . . 5 (𝑤 = 𝐵 → ((𝐴𝑤𝑤𝐴) ↔ (𝐴𝐵𝐵𝐴)))
12 breq2 5069 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1311, 12imbi12d 347 . . . 4 (𝑤 = 𝐵 → (((𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
14 vex 3497 . . . . 5 𝑧 ∈ V
15 sseq1 3991 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
16 imaeq2 5924 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1716difeq2d 4098 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1817imaeq2d 5928 . . . . . . . 8 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
19 difeq2 4092 . . . . . . . 8 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2018, 19sseq12d 3999 . . . . . . 7 (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥)))
2115, 20anbi12d 632 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦)) ↔ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))))
2221cbvabv 2889 . . . . 5 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
23 eqid 2821 . . . . 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 3497 . . . . 5 𝑤 ∈ V
2514, 22, 23, 24sbthlem10 8635 . . . 4 ((𝑧𝑤𝑤𝑧) → 𝑧𝑤)
268, 13, 25vtocl2g 3571 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
272, 3, 26syl2an 597 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
2827pm2.43i 52 1 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {cab 2799  Vcvv 3494  cdif 3932  cun 3933  wss 3935   cuni 4837   class class class wbr 5065  ccnv 5553  cres 5556  cima 5557  cen 8505  cdom 8506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-en 8509  df-dom 8510
This theorem is referenced by:  sbthb  8637  sdomnsym  8641  domtriord  8662  xpen  8679  limenpsi  8691  php  8700  onomeneq  8707  unbnn  8773  infxpenlem  9438  fseqen  9452  infpwfien  9487  inffien  9488  alephdom  9506  mappwen  9537  infdjuabs  9627  infunabs  9628  infdju  9629  infdif  9630  infxpabs  9633  infmap2  9639  gchaleph  10092  gchhar  10100  inttsk  10195  inar1  10196  znnen  15564  qnnen  15565  rpnnen  15579  rexpen  15580  mreexfidimd  16920  acsinfdimd  17791  fislw  18749  opnreen  23438  ovolctb2  24092  vitali  24213  aannenlem3  24918  basellem4  25660  lgsqrlem4  25924  upgrex  26876  ctbssinf  34686  phpreu  34875  poimirlem26  34917  pellexlem4  39429  pellexlem5  39430  idomsubgmo  39798
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