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Theorem frpoind 6363
Description: The principle of well-founded induction over a partial order. This theorem is a version of frind 9790 that does not require the axiom of infinity and can be used to prove wfi 6371 and tfi 7874. (Contributed by Scott Fenton, 11-Feb-2022.)
Assertion
Ref Expression
frpoind (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem frpoind
StepHypRef Expression
1 ssdif0 4366 . . . . . . 7 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2988 . . . . . 6 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 difss 4136 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
4 frpomin2 6362 . . . . . . . . 9 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)
5 eldif 3961 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
65anbi1i 624 . . . . . . . . . . . 12 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ ((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅))
7 anass 468 . . . . . . . . . . . 12 (((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)))
8 indif2 4281 . . . . . . . . . . . . . . . . 17 ((𝑅 “ {𝑦}) ∩ (𝐴𝐵)) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
9 df-pred 6321 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝐴𝐵) ∩ (𝑅 “ {𝑦}))
10 incom 4209 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
119, 10eqtri 2765 . . . . . . . . . . . . . . . . 17 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
12 df-pred 6321 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (𝑅 “ {𝑦}))
13 incom 4209 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1412, 13eqtri 2765 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, 𝐴, 𝑦) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1514difeq1i 4122 . . . . . . . . . . . . . . . . 17 (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
168, 11, 153eqtr4i 2775 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
1716eqeq1i 2742 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
18 ssdif0 4366 . . . . . . . . . . . . . . 15 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
1917, 18bitr4i 278 . . . . . . . . . . . . . 14 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
2019anbi1ci 626 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
2120anbi2i 623 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
226, 7, 213bitri 297 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
2322rexbii2 3090 . . . . . . . . . 10 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
24 rexanali 3102 . . . . . . . . . 10 (∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2523, 24bitri 275 . . . . . . . . 9 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
264, 25sylib 218 . . . . . . . 8 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2726ex 412 . . . . . . 7 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
283, 27mpani 696 . . . . . 6 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ((𝐴𝐵) ≠ ∅ → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
292, 28biimtrid 242 . . . . 5 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (¬ 𝐴𝐵 → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
3029con4d 115 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵) → 𝐴𝐵))
3130imp 406 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴𝐵)
3231adantrl 716 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴𝐵)
33 simprl 771 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐵𝐴)
3432, 33eqssd 4001 1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333  {csn 4626   Po wpo 5590   Fr wfr 5634   Se wse 5635  ccnv 5684  cima 5688  Predcpred 6320
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-fr 5637  df-se 5638  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  frpoinsg  6364  wfi  6371
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