Theorem sbth 3359

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 3349 through sbthlem10 3358; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 3358. We closely follow the proof in Suppes, which you should consult to understand our proof at a higher level.

Assertion

Ref	Expression
sbth	|- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)

Proof of Theorem sbth

Step	Hyp	Ref	Expression
1		breq1 2065	. . . . . 6 |- (z = A -> (z ~<_ w <-> A ~<_ w))
2		breq2 2066	. . . . . 6 |- (z = A -> (w ~<_ z <-> w ~<_ A))
3	1, 2	anbi12d 476	. . . . 5 |- (z = A -> ((z ~<_ w /\ w ~<_ z) <-> (A ~<_ w /\ w ~<_ A)))
4		breq1 2065	. . . . 5 |- (z = A -> (z ~~ w <-> A ~~ w))
5	3, 4	imbi12d 474	. . . 4 |- (z = A -> (((z ~<_ w /\ w ~<_ z) -> z ~~ w) <-> ((A ~<_ w /\ w ~<_ A) -> A ~~ w)))
6		breq2 2066	. . . . . 6 |- (w = B -> (A ~<_ w <-> A ~<_ B))
7		breq1 2065	. . . . . 6 |- (w = B -> (w ~<_ A <-> B ~<_ A))
8	6, 7	anbi12d 476	. . . . 5 |- (w = B -> ((A ~<_ w /\ w ~<_ A) <-> (A ~<_ B /\ B ~<_ A)))
9		breq2 2066	. . . . 5 |- (w = B -> (A ~~ w <-> A ~~ B))
10	8, 9	imbi12d 474	. . . 4 |- (w = B -> (((A ~<_ w /\ w ~<_ A) -> A ~~ w) <-> ((A ~<_ B /\ B ~<_ A) -> A ~~ B)))
11		visset 1350	. . . . 5 |- z e. V
12		sseq1 1521	. . . . . . 7 |- (y = x -> (y (_ z <-> x (_ z))
13		imaeq2 2603	. . . . . . . . . 10 |- (y = x -> (f"y) = (f"x))
14	13	difeq2d 1588	. . . . . . . . 9 |- (y = x -> (w \ (f"y)) = (w \ (f"x)))
15		imaeq2 2603	. . . . . . . . 9 |- ((w \ (f"y)) = (w \ (f"x)) -> (g"(w \ (f"y))) = (g"(w \ (f"x))))
16		sseq1 1521	. . . . . . . . 9 |- ((g"(w \ (f"y))) = (g"(w \ (f"x))) -> ((g"(w \ (f"y))) (_ (z \ y) <-> (g"(w \ (f"x))) (_ (z \ y)))
17	14, 15, 16	3syl 21	. . . . . . . 8 |- (y = x -> ((g"(w \ (f"y))) (_ (z \ y) <-> (g"(w \ (f"x))) (_ (z \ y)))
18		difeq2 1583	. . . . . . . . 9 |- (y = x -> (z \ y) = (z \ x))
19	18	sseq2d 1528	. . . . . . . 8 |- (y = x -> ((g"(w \ (f"x))) (_ (z \ y) <-> (g"(w \ (f"x))) (_ (z \ x)))
20	17, 19	bitrd 406	. . . . . . 7 |- (y = x -> ((g"(w \ (f"y))) (_ (z \ y) <-> (g"(w \ (f"x))) (_ (z \ x)))
21	12, 20	anbi12d 476	. . . . . 6 |- (y = x -> ((y (_ z /\ (g"(w \ (f"y))) (_ (z \ y)) <-> (x (_ z /\ (g"(w \ (f"x))) (_ (z \ x))))
22	21	cbvabv 1424	. . . . 5 |- {y | (y (_ z /\ (g"(w \ (f"y))) (_ (z \ y))} = {x | (x (_ z /\ (g"(w \ (f"x))) (_ (z \ x))}
23		cleqid 1102	. . . . 5 |- ((f |` U.{y | (y (_ z /\ (g"(w \ (f"y))) (_ (z \ y))}) u. (`'g |` (z \ U.{y | (y (_ z /\ (g"(w \ (f"y))) (_ (z \ y))}))) = ((f |` U.{y | (y (_ z /\ (g"(w \ (f"y))) (_ (z \ y))}) u. (`'g |` (z \ U.{y | (y (_ z /\ (g"(w \ (f"y))) (_ (z \ y))})))
24		visset 1350	. . . . 5 |- w e. V
25	11, 22, 23, 24	sbthlem10 3358	. . . 4 |- ((z ~<_ w /\ w ~<_ z) -> z ~~ w)
26	5, 10, 25	vtocl2g 1386	. . 3 |- ((A e. V /\ B e. V) -> ((A ~<_ B /\ B ~<_ A) -> A ~~ B))
27		reldom 3278	. . . 4 |- Rel ~<_
28	27	brrelexi 2447	. . 3 |- (A ~<_ B -> A e. V)
29	27	brrelexi 2447	. . 3 |- (B ~<_ A -> B e. V)
30	26, 28, 29	syl2an 349	. 2 |- ((A ~<_ B /\ B ~<_ A) -> ((A ~<_ B /\ B ~<_ A) -> A ~~ B))
31	30	pm2.43i 58	1 |- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   (_ wss 1487  U.cuni 1919   class class class wbr 2054  `'ccnv 2409   |` cres 2412  "cima 2413   ~~ cen 3271   ~<_ cdom 3272

This theorem is referenced by:  sbthbg 3360  sdomnsym 3364  sdomdomtr 3370  limenpsi 3400  php 3409  onomeneq 3414  unbnn 3435  xpnnen 4927  znnen 4930  qnnen 4931  infxpidmlem1 4933  infxpidmlem12 4944  infunabs 4946  infcdaabs 4947  infdif 4948  infxpabs 4949  infmap1 4950  infmap2 4953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274  df-dom 3275

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