Theorem isbth 7005

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 6995 through sbthlemi10 7004; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7004. 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 15312. (Contributed by NM, 8-Jun-1998.)

Assertion

Ref	Expression
isbth	⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)

Proof of Theorem isbth

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

Step	Hyp	Ref	Expression
1		simprl 529	. 2 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≼ 𝐵)
2		simprr 531	. 2 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐵 ≼ 𝐴)
3		reldom 6779	. . . . 5 ⊢ Rel ≼
4	3	brrelex1i 4694	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
5	2, 4	syl 14	. . 3 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐵 ∈ V)
6		breq2 4029	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵))
7		breq1 4028	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴))
8	6, 7	anbi12d 473	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
9		breq2 4029	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵))
10	8, 9	imbi12d 234	. . . 4 ⊢ (𝑤 = 𝐵 → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)))
11	10	adantl 277	. . 3 ⊢ (((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) ∧ 𝑤 = 𝐵) → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)))
12	3	brrelex1i 4694	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
13	1, 12	syl 14	. . . 4 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ∈ V)
14		breq1 4028	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤))
15		breq2 4029	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴))
16	14, 15	anbi12d 473	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴)))
17		breq1 4028	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤))
18	16, 17	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤)))
19	18	adantl 277	. . . 4 ⊢ (((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) ∧ 𝑧 = 𝐴) → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤)))
20		vex 2759	. . . . . . 7 ⊢ 𝑧 ∈ V
21		sseq1 3197	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧))
22		imaeq2 4991	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥))
23	22	difeq2d 3272	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥)))
24	23	imaeq2d 4995	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))))
25		difeq2 3266	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥))
26	24, 25	sseq12d 3205	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))
27	21, 26	anbi12d 473	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))))
28	27	cbvabv 2314	. . . . . . 7 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))}
29		eqid 2189	. . . . . . 7 ⊢ ((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))})))
30		vex 2759	. . . . . . 7 ⊢ 𝑤 ∈ V
31	20, 28, 29, 30	sbthlemi10 7004	. . . . . 6 ⊢ ((EXMID ∧ (𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧)) → 𝑧 ≈ 𝑤)
32	31	ex 115	. . . . 5 ⊢ (EXMID → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤))
33	32	adantr 276	. . . 4 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤))
34	13, 19, 33	vtocld 2808	. . 3 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤))
35	5, 11, 34	vtocld 2808	. 2 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
36	1, 2, 35	mp2and 433	1 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  {cab 2175  Vcvv 2756   ∖ cdif 3145   ∪ cun 3146   ⊆ wss 3148  ∪ cuni 3831   class class class wbr 4025  EXMIDwem 4219  ^◡ccnv 4650   ↾ cres 4653   “ cima 4654   ≈ cen 6772   ≼ cdom 6773

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-exmid 4220  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-en 6775  df-dom 6776

This theorem is referenced by:  exmidsbth  15313