Theorem sbth 9124

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 9114 through sbthlem10 9123; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 9123. 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 9123. 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 8987	. . . 4 ⊢ Rel ≼
2	1	brrelex1i 5707	. . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	1	brrelex1i 5707	. . 3 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
4		breq1 5105	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤))
5		breq2 5106	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴))
6	4, 5	anbi12d 644	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴)))
7		breq1 5105	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤))
8	6, 7	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝐴 → (((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤)))
9		breq2 5106	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵))
10		breq1 5105	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴))
11	9, 10	anbi12d 644	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
12		breq2 5106	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵))
13	11, 12	imbi12d 347	. . . 4 ⊢ (𝑤 = 𝐵 → (((𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)))
14		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
15		sseq1 3934	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧))
16		imaeq2 6050	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥))
17	16	difeq2d 4062	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥)))
18	17	imaeq2d 6054	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))))
19		difeq2 4056	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥))
20	18, 19	sseq12d 3942	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))
21	15, 20	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))))
22	21	cbvabv 2830	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))}
23		eqid 2760	. . . . 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 3454	. . . . 5 ⊢ 𝑤 ∈ V
25	14, 22, 23, 24	sbthlem10 9123	. . . 4 ⊢ ((𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤)
26	8, 13, 25	vtocl2g 3533	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
27	2, 3, 26	syl2an 608	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
28	27	pm2.43i 53	1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   ∪ cun 4148  ∪ cuni 4866   class class class wbr 5102  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654   ≈ cen 8978   ≼ cdom 8979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-en 8982  df-dom 8983

This theorem is used by:  sbthb  9125  sdomnsym  9129  domtriord  9150  xpen  9167  limenpsi  9179  unbnn  9296  infxpenlem  10099  fseqen  10113  infpwfien  10148  inffien  10149  alephdom  10167  mappwen  10198  infdjuabs  10290  infunabs  10291  infdju  10292  infdif  10293  infxpabs  10296  infmap2  10302  gchaleph  10763  gchhar  10771  inttsk  10866  inar1  10867  znnen  16389  qnnen  16390  rpnnen  16404  rexpen  16405  mreexfidimd  17833  acsinfdimd  18741  fislw  19848  opnreen  25160  ovolctb2  25822  vitali  25943  aannenlem3  26666  basellem4  27420  lgsqrlem4  27685  upgrex  29679  weexenwe  35824  iccioo01  38387  ctbssinf  38466  phpreu  38664  poimirlem26  38701  pellexlem4  43976  pellexlem5  43977  idomsubgmo  44332