Theorem sbthfi 9140

Description: Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9038). (Contributed by BTernaryTau, 4-Nov-2024.)

Assertion

Ref	Expression
sbthfi	⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Proof of Theorem sbthfi

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

Step	Hyp	Ref	Expression
1		reldom 8901	. . . . 5 ⊢ Rel ≼
2	1	brrelex1i 5687	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	1	brrelex1i 5687	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
4		breq1 5105	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤))
5		breq2 5106	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑤 ≼ 𝑧 ↔ 𝑤 ≼ 𝐴))
6	4, 5	3anbi23d 1441	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑤 ∈ Fin ∧ 𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) ↔ (𝑤 ∈ Fin ∧ 𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴)))
7		breq1 5105	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤))
8	6, 7	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑤 ∈ Fin ∧ 𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤) ↔ ((𝑤 ∈ Fin ∧ 𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤)))
9		eleq1 2816	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ Fin ↔ 𝐵 ∈ Fin))
10		breq2 5106	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵))
11		breq1 5105	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ≼ 𝐴 ↔ 𝐵 ≼ 𝐴))
12	9, 10, 11	3anbi123d 1438	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑤 ∈ Fin ∧ 𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) ↔ (𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
13		breq2 5106	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ≈ 𝑤 ↔ 𝐴 ≈ 𝐵))
14	12, 13	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐵 → (((𝑤 ∈ Fin ∧ 𝐴 ≼ 𝑤 ∧ 𝑤 ≼ 𝐴) → 𝐴 ≈ 𝑤) ↔ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)))
15		vex 3448	. . . . . 6 ⊢ 𝑧 ∈ V
16		sseq1 3969	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧))
17		imaeq2 6016	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥))
18	17	difeq2d 4085	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥)))
19	18	imaeq2d 6020	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))))
20		difeq2 4079	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥))
21	19, 20	sseq12d 3977	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥)))
22	16, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦)) ↔ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))))
23	22	cbvabv 2799	. . . . . 6 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))}
24		eqid 2729	. . . . . 6 ⊢ ((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))})))
25		vex 3448	. . . . . 6 ⊢ 𝑤 ∈ V
26	15, 23, 24, 25	sbthfilem 9139	. . . . 5 ⊢ ((𝑤 ∈ Fin ∧ 𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤)
27	8, 14, 26	vtocl2g 3537	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
28	2, 3, 27	syl2an 596	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
29	28	3adant1 1130	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵))
30	29	pm2.43i 52	1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  ∪ cuni 4867   class class class wbr 5102  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-dom 8897  df-fin 8899

This theorem is referenced by:  domnsymfi  9141  php  9148