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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. 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 9143 through sbthlem10 9152; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 9152. 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 9152. 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 9003 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5751 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | 1 | brrelex1i 5751 | . . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V) |
4 | breq1 5169 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤)) | |
5 | breq2 5170 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴)) | |
6 | 4, 5 | anbi12d 631 | . . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴))) |
7 | breq1 5169 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤)) | |
8 | 6, 7 | imbi12d 344 | . . . 4 ⊢ (𝑧 = 𝐴 → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤))) |
9 | breq2 5170 | . . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵)) | |
10 | breq1 5169 | . . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴)) | |
11 | 9, 10 | anbi12d 631 | . . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴))) |
12 | breq2 5170 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵)) | |
13 | 11, 12 | imbi12d 344 | . . . 4 ⊢ (𝑤 = 𝐵 → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))) |
14 | vex 3492 | . . . . 5 ⊢ 𝑧 ∈ V | |
15 | sseq1 4034 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧)) | |
16 | imaeq2 6080 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥)) | |
17 | 16 | difeq2d 4149 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥))) |
18 | 17 | imaeq2d 6084 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥)))) |
19 | difeq2 4143 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥)) | |
20 | 18, 19 | sseq12d 4042 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))) |
21 | 15, 20 | anbi12d 631 | . . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))) |
22 | 21 | cbvabv 2815 | . . . . 5 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))} |
23 | eqid 2740 | . . . . 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 3492 | . . . . 5 ⊢ 𝑤 ∈ V | |
25 | 14, 22, 23, 24 | sbthlem10 9152 | . . . 4 ⊢ ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) |
26 | 8, 13, 25 | vtocl2g 3586 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)) |
27 | 2, 3, 26 | syl2an 595 | . 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 395 = wceq 1537 ∈ wcel 2108 {cab 2717 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 ∪ cuni 4931 class class class wbr 5166 ◡ccnv 5694 ↾ cres 5697 “ cima 5698 ≈ cen 8994 ≼ cdom 8995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7764 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5701 df-rel 5702 df-cnv 5703 df-co 5704 df-dm 5705 df-rn 5706 df-res 5707 df-ima 5708 df-fun 6570 df-fn 6571 df-f 6572 df-f1 6573 df-fo 6574 df-f1o 6575 df-en 8998 df-dom 8999 |
This theorem is referenced by: sbthb 9154 sdomnsym 9158 domtriord 9183 xpen 9200 limenpsi 9212 phpOLD 9279 onomeneqOLD 9286 unbnn 9354 infxpenlem 10076 fseqen 10090 infpwfien 10125 inffien 10126 alephdom 10144 mappwen 10175 infdjuabs 10268 infunabs 10269 infdju 10270 infdif 10271 infxpabs 10274 infmap2 10280 gchaleph 10734 gchhar 10742 inttsk 10837 inar1 10838 znnen 16254 qnnen 16255 rpnnen 16269 rexpen 16270 mreexfidimd 17702 acsinfdimd 18622 fislw 19661 opnreen 24864 ovolctb2 25538 vitali 25659 aannenlem3 26382 basellem4 27137 lgsqrlem4 27403 upgrex 29119 iccioo01 37286 ctbssinf 37365 phpreu 37557 poimirlem26 37599 pellexlem4 42781 pellexlem5 42782 idomsubgmo 43150 |
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