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Theorem frpoind 32205
Description: The principle of founded induction over a partial ordering. This theorem is a version of frind 32208 that does not require infinity, and can be used to prove wfi 5900 and tfi 7255. (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 4108 . . . . . . 7 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2984 . . . . . 6 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 difss 3901 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
4 frpomin2 32204 . . . . . . . . 9 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)
5 eldif 3744 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
65anbi1i 617 . . . . . . . . . . . 12 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ ((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅))
7 anass 460 . . . . . . . . . . . 12 (((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)))
8 indif2 4037 . . . . . . . . . . . . . . . . . 18 ((𝑅 “ {𝑦}) ∩ (𝐴𝐵)) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
9 df-pred 5867 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝐴𝐵) ∩ (𝑅 “ {𝑦}))
10 incom 3969 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
119, 10eqtri 2787 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
12 df-pred 5867 . . . . . . . . . . . . . . . . . . . 20 Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (𝑅 “ {𝑦}))
13 incom 3969 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1412, 13eqtri 2787 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, 𝐴, 𝑦) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1514difeq1i 3888 . . . . . . . . . . . . . . . . . 18 (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
168, 11, 153eqtr4i 2797 . . . . . . . . . . . . . . . . 17 Pred(𝑅, (𝐴𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
1716eqeq1i 2770 . . . . . . . . . . . . . . . 16 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
18 ssdif0 4108 . . . . . . . . . . . . . . . 16 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
1917, 18bitr4i 269 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
2019anbi2i 616 . . . . . . . . . . . . . 14 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (¬ 𝑦𝐵 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
21 ancom 452 . . . . . . . . . . . . . 14 ((¬ 𝑦𝐵 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
2220, 21bitri 266 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
2322anbi2i 616 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
246, 7, 233bitri 288 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
2524rexbii2 3186 . . . . . . . . . 10 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
26 rexanali 3144 . . . . . . . . . 10 (∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2725, 26bitri 266 . . . . . . . . 9 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
284, 27sylib 209 . . . . . . . 8 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2928ex 401 . . . . . . 7 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
303, 29mpani 687 . . . . . 6 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ((𝐴𝐵) ≠ ∅ → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
312, 30syl5bi 233 . . . . 5 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (¬ 𝐴𝐵 → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
3231con4d 115 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵) → 𝐴𝐵))
3332imp 395 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴𝐵)
3433adantrl 707 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴𝐵)
35 simprl 787 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐵𝐴)
3634, 35eqssd 3780 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 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  cdif 3731  cin 3733  wss 3734  c0 4081  {csn 4336   Po wpo 5198   Fr wfr 5235   Se wse 5236  ccnv 5278  cima 5282  Predcpred 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-po 5200  df-fr 5238  df-se 5239  df-xp 5285  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867
This theorem is referenced by:  frpoinsg  32206
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