Theorem sbth 7965

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 7955 through sbthlem10 7964; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7964. 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	⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Proof of Theorem sbth

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7847	. . . 4 ⊢ Rel ≼
2	1	brrelexi 5082	. . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	1	brrelexi 5082	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
4		breq1 4586	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤))
5		breq2 4587	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴))
6	4, 5	anbi12d 743	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴)))
7		breq1 4586	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤))
8	6, 7	imbi12d 333	. . . 4 ⊢ (𝑧 = 𝐴 → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤)))
9		breq2 4587	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵))
10		breq1 4586	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴))
11	9, 10	anbi12d 743	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
12		breq2 4587	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵))
13	11, 12	imbi12d 333	. . . 4 ⊢ (𝑤 = 𝐵 → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)))
14		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
15		sseq1 3589	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧))
16		imaeq2 5381	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥))
17	16	difeq2d 3690	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥)))
18	17	imaeq2d 5385	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))))
19		difeq2 3684	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥))
20	18, 19	sseq12d 3597	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))
21	15, 20	anbi12d 743	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))))
22	21	cbvabv 2734	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))}
23		eqid 2610	. . . . 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 3176	. . . . 5 ⊢ 𝑤 ∈ V
25	14, 22, 23, 24	sbthlem10 7964	. . . 4 ⊢ ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤)
26	8, 13, 25	vtocl2g 3243	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
27	2, 3, 26	syl2an 493	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
28	27	pm2.43i 50	1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  ^◡ccnv 5037   ↾ cres 5040   “ cima 5041   ≈ cen 7838   ≼ cdom 7839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-en 7842  df-dom 7843

This theorem is referenced by:  sbthb  7966  sdomnsym  7970  domtriord  7991  xpen  8008  limenpsi  8020  php  8029  onomeneq  8035  unbnn  8101  infxpenlem  8719  fseqen  8733  infpwfien  8768  inffien  8769  alephdom  8787  mappwen  8818  infcdaabs  8911  infunabs  8912  infcda  8913  infdif  8914  infxpabs  8917  infmap2  8923  gchaleph  9372  gchhar  9380  inttsk  9475  inar1  9476  znnen  14780  qnnen  14781  rpnnen  14795  rexpen  14796  mreexfidimd  16134  acsinfdimd  17005  fislw  17863  opnreen  22442  ovolctb2  23067  vitali  23188  aannenlem3  23889  basellem4  24610  lgsqrlem4  24874  upgrex  25759  umgraex  25852  phpreu  32563  poimirlem26  32605  pellexlem4  36414  pellexlem5  36415  idomsubgmo  36795