Theorem frpoind 6331

Description: The principle of well-founded induction over a partial order. This theorem is a version of frind 9710 that does not require the axiom of infinity and can be used to prove wfi 6338 and tfi 7835. (Contributed by Scott Fenton, 11-Feb-2022.)

Assertion

Ref	Expression
frpoind	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem frpoind

Step	Hyp	Ref	Expression
1		ssdif0 4321	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 3006	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		difss 4091	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
4		frpomin2 6330	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)
5		eldif 3916	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
6	5	anbi1i 633	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅))
7		anass 472	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)))
8		indif2 4235	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵)) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
9		df-pred 6290	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦}))
10		incom 4163	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
11	9, 10	eqtri 2787	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
12		df-pred 6290	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (◡𝑅 “ {𝑦}))
13		incom 4163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
14	12, 13	eqtri 2787	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, 𝐴, 𝑦) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
15	14	difeq1i 4078	. . . . . . . . . . . . . . . . 17 ⊢ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
16	8, 11, 15	3eqtr4i 2797	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
17	16	eqeq1i 2769	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
18		ssdif0 4321	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19	17, 18	bitr4i 280	. . . . . . . . . . . . . 14 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
20	19	anbi1ci 635	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
21	20	anbi2i 632	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
22	6, 7, 21	3bitri 299	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
23	22	rexbii2 3107	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
24		rexanali 3118	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
25	23, 24	bitri 277	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
26	4, 25	sylib 220	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
27	26	ex 416	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
28	3, 27	mpani 706	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ((𝐴 ∖ 𝐵) ≠ ∅ → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
29	2, 28	biimtrid 244	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (¬ 𝐴 ⊆ 𝐵 → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
30	29	con4d 115	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵) → 𝐴 ⊆ 𝐵))
31	30	imp 410	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
32	31	adantrl 726	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 ⊆ 𝐵)
33		simprl 780	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐵 ⊆ 𝐴)
34	32, 33	eqssd 3955	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4584   Po wpo 5555   Fr wfr 5599   Se wse 5600  ^◡ccnv 5648   “ cima 5652  Predcpred 6289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-po 5557  df-fr 5602  df-se 5603  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290

This theorem is referenced by:  frpoinsg  6332  wfi  6338