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Mirrors > Home > MPE Home > Th. List > sbth | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sbth | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8247 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5406 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 1 | brrelex1i 5406 | . . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V) |
4 | breq1 4889 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤)) | |
5 | breq2 4890 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴)) | |
6 | 4, 5 | anbi12d 624 | . . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴))) |
7 | breq1 4889 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤)) | |
8 | 6, 7 | imbi12d 336 | . . . 4 ⊢ (𝑧 = 𝐴 → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤))) |
9 | breq2 4890 | . . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵)) | |
10 | breq1 4889 | . . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴)) | |
11 | 9, 10 | anbi12d 624 | . . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴))) |
12 | breq2 4890 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵)) | |
13 | 11, 12 | imbi12d 336 | . . . 4 ⊢ (𝑤 = 𝐵 → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))) |
14 | vex 3400 | . . . . 5 ⊢ 𝑧 ∈ V | |
15 | sseq1 3844 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧)) | |
16 | imaeq2 5716 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥)) | |
17 | 16 | difeq2d 3950 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥))) |
18 | 17 | imaeq2d 5720 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥)))) |
19 | difeq2 3944 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥)) | |
20 | 18, 19 | sseq12d 3852 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))) |
21 | 15, 20 | anbi12d 624 | . . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))) |
22 | 21 | cbvabv 2913 | . . . . 5 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))} |
23 | eqid 2777 | . . . . 5 ⊢ ((𝑓 ↾ ∪ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))}) ∪ (◡𝑔 ↾ (𝑧 ∖ ∪ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))}))) = ((𝑓 ↾ ∪ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))}) ∪ (◡𝑔 ↾ (𝑧 ∖ ∪ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))}))) | |
24 | vex 3400 | . . . . 5 ⊢ 𝑤 ∈ V | |
25 | 14, 22, 23, 24 | sbthlem10 8367 | . . . 4 ⊢ ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) |
26 | 8, 13, 25 | vtocl2g 3470 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)) |
27 | 2, 3, 26 | syl2an 589 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)) |
28 | 27 | pm2.43i 52 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
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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