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Theorem isbth 6966
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 6956 through sbthlemi10 6965; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6965. 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. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 14774. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
isbth ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)

Proof of Theorem isbth
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . 2 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)
2 simprr 531 . 2 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐵𝐴)
3 reldom 6745 . . . . 5 Rel ≼
43brrelex1i 4670 . . . 4 (𝐵𝐴𝐵 ∈ V)
52, 4syl 14 . . 3 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐵 ∈ V)
6 breq2 4008 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
7 breq1 4007 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
86, 7anbi12d 473 . . . . 5 (𝑤 = 𝐵 → ((𝐴𝑤𝑤𝐴) ↔ (𝐴𝐵𝐵𝐴)))
9 breq2 4008 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
108, 9imbi12d 234 . . . 4 (𝑤 = 𝐵 → (((𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
1110adantl 277 . . 3 (((EXMID ∧ (𝐴𝐵𝐵𝐴)) ∧ 𝑤 = 𝐵) → (((𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
123brrelex1i 4670 . . . . 5 (𝐴𝐵𝐴 ∈ V)
131, 12syl 14 . . . 4 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴 ∈ V)
14 breq1 4007 . . . . . . 7 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
15 breq2 4008 . . . . . . 7 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
1614, 15anbi12d 473 . . . . . 6 (𝑧 = 𝐴 → ((𝑧𝑤𝑤𝑧) ↔ (𝐴𝑤𝑤𝐴)))
17 breq1 4007 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
1816, 17imbi12d 234 . . . . 5 (𝑧 = 𝐴 → (((𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
1918adantl 277 . . . 4 (((EXMID ∧ (𝐴𝐵𝐵𝐴)) ∧ 𝑧 = 𝐴) → (((𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
20 vex 2741 . . . . . . 7 𝑧 ∈ V
21 sseq1 3179 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
22 imaeq2 4967 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
2322difeq2d 3254 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
2423imaeq2d 4971 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
25 difeq2 3248 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2624, 25sseq12d 3187 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥)))
2721, 26anbi12d 473 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦)) ↔ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))))
2827cbvabv 2302 . . . . . . 7 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
29 eqid 2177 . . . . . . 7 ((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))})))
30 vex 2741 . . . . . . 7 𝑤 ∈ V
3120, 28, 29, 30sbthlemi10 6965 . . . . . 6 ((EXMID ∧ (𝑧𝑤𝑤𝑧)) → 𝑧𝑤)
3231ex 115 . . . . 5 (EXMID → ((𝑧𝑤𝑤𝑧) → 𝑧𝑤))
3332adantr 276 . . . 4 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → ((𝑧𝑤𝑤𝑧) → 𝑧𝑤))
3413, 19, 33vtocld 2790 . . 3 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → ((𝐴𝑤𝑤𝐴) → 𝐴𝑤))
355, 11, 34vtocld 2790 . 2 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → ((𝐴𝐵𝐵𝐴) → 𝐴𝐵))
361, 2, 35mp2and 433 1 ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  Vcvv 2738  cdif 3127  cun 3128  wss 3130   cuni 3810   class class class wbr 4004  EXMIDwem 4195  ccnv 4626  cres 4629  cima 4630  cen 6738  cdom 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-exmid 4196  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-en 6741  df-dom 6742
This theorem is referenced by:  exmidsbth  14775
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