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Theorem frpoind 6297
Description: The principle of well-founded induction over a partial order. This theorem is a version of frind 9691 that does not require the axiom of infinity and can be used to prove wfi 6305 and tfi 7790. (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 4324 . . . . . . 7 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2988 . . . . . 6 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 difss 4092 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
4 frpomin2 6296 . . . . . . . . 9 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)
5 eldif 3921 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
65anbi1i 625 . . . . . . . . . . . 12 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ ((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅))
7 anass 470 . . . . . . . . . . . 12 (((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)))
8 indif2 4231 . . . . . . . . . . . . . . . . 17 ((𝑅 “ {𝑦}) ∩ (𝐴𝐵)) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
9 df-pred 6254 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝐴𝐵) ∩ (𝑅 “ {𝑦}))
10 incom 4162 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
119, 10eqtri 2761 . . . . . . . . . . . . . . . . 17 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
12 df-pred 6254 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (𝑅 “ {𝑦}))
13 incom 4162 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1412, 13eqtri 2761 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, 𝐴, 𝑦) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1514difeq1i 4079 . . . . . . . . . . . . . . . . 17 (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
168, 11, 153eqtr4i 2771 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
1716eqeq1i 2738 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
18 ssdif0 4324 . . . . . . . . . . . . . . 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 627 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
2120anbi2i 624 . . . . . . . . . . . 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 217 . . . . . . . 8 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2726ex 414 . . . . . . 7 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
283, 27mpani 695 . . . . . 6 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ((𝐴𝐵) ≠ ∅ → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
292, 28biimtrid 241 . . . . 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 408 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴𝐵)
3231adantrl 715 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴𝐵)
33 simprl 770 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐵𝐴)
3432, 33eqssd 3962 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  cdif 3908  cin 3910  wss 3911  c0 4283  {csn 4587   Po wpo 5544   Fr wfr 5586   Se wse 5587  ccnv 5633  cima 5637  Predcpred 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-po 5546  df-fr 5589  df-se 5590  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  frpoinsg  6298  wfi  6305
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