Theorem sbthfi 9239

Description: Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9133). (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 8991	. . . . 5 ⊢ Rel ≼
2	1	brrelex1i 5741	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	1	brrelex1i 5741	. . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
4		breq1 5146	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 ≼ 𝑤 ↔ 𝐴 ≼ 𝑤))
5		breq2 5147	. . . . . . 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 5146	. . . . . 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 2829	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ Fin ↔ 𝐵 ∈ Fin))
10		breq2 5147	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝐴 ≼ 𝑤 ↔ 𝐴 ≼ 𝐵))
11		breq1 5146	. . . . . . 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 5147	. . . . . 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 3484	. . . . . 6 ⊢ 𝑧 ∈ V
16		sseq1 4009	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑧 ↔ 𝑥 ⊆ 𝑧))
17		imaeq2 6074	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑓 “ 𝑦) = (𝑓 “ 𝑥))
18	17	difeq2d 4126	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑤 ∖ (𝑓 “ 𝑦)) = (𝑤 ∖ (𝑓 “ 𝑥)))
19	18	imaeq2d 6078	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))))
20		difeq2 4120	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑧 ∖ 𝑦) = (𝑧 ∖ 𝑥))
21	19, 20	sseq12d 4017	. . . . . . . 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 2812	. . . . . 6 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑦))) ⊆ (𝑧 ∖ 𝑦))} = {𝑥 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓 “ 𝑥))) ⊆ (𝑧 ∖ 𝑥))}
24		eqid 2737	. . . . . 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 3484	. . . . . 6 ⊢ 𝑤 ∈ V
26	15, 23, 24, 25	sbthfilem 9238	. . . . 5 ⊢ ((𝑤 ∈ Fin ∧ 𝑧 ≼ 𝑤 ∧ 𝑤 ≼ 𝑧) → 𝑧 ≈ 𝑤)
27	8, 14, 26	vtocl2g 3574	. . . 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 1131	. 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 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∪ cuni 4907   class class class wbr 5143  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688   ≈ cen 8982   ≼ cdom 8983  Fincfn 8985

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989

This theorem is referenced by:  domnsymfi  9240  php  9247