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Theorem wfi 5856
Description: The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
wfi (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem wfi
StepHypRef Expression
1 ssdif0 4089 . . . . . . 7 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2990 . . . . . 6 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 difss 3888 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
4 tz6.26 5854 . . . . . . . . 9 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)
5 eldif 3733 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
65anbi1i 610 . . . . . . . . . . . 12 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ ((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅))
7 anass 459 . . . . . . . . . . . 12 (((𝑦𝐴 ∧ ¬ 𝑦𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)))
8 ancom 452 . . . . . . . . . . . . . 14 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ∧ ¬ 𝑦𝐵))
9 indif2 4019 . . . . . . . . . . . . . . . . . 18 ((𝑅 “ {𝑦}) ∩ (𝐴𝐵)) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
10 df-pred 5823 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝐴𝐵) ∩ (𝑅 “ {𝑦}))
11 incom 3956 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
1210, 11eqtri 2793 . . . . . . . . . . . . . . . . . 18 Pred(𝑅, (𝐴𝐵), 𝑦) = ((𝑅 “ {𝑦}) ∩ (𝐴𝐵))
13 df-pred 5823 . . . . . . . . . . . . . . . . . . . 20 Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (𝑅 “ {𝑦}))
14 incom 3956 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∩ (𝑅 “ {𝑦})) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1513, 14eqtri 2793 . . . . . . . . . . . . . . . . . . 19 Pred(𝑅, 𝐴, 𝑦) = ((𝑅 “ {𝑦}) ∩ 𝐴)
1615difeq1i 3875 . . . . . . . . . . . . . . . . . 18 (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
179, 12, 163eqtr4i 2803 . . . . . . . . . . . . . . . . 17 Pred(𝑅, (𝐴𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
1817eqeq1i 2776 . . . . . . . . . . . . . . . 16 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19 ssdif0 4089 . . . . . . . . . . . . . . . 16 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
2018, 19bitr4i 267 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
2120anbi1i 610 . . . . . . . . . . . . . 14 ((Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ∧ ¬ 𝑦𝐵) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
228, 21bitri 264 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
2322anbi2i 609 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (¬ 𝑦𝐵 ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅)) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
246, 7, 233bitri 286 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴𝐵) ∧ Pred(𝑅, (𝐴𝐵), 𝑦) = ∅) ↔ (𝑦𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵)))
2524rexbii2 3187 . . . . . . . . . 10 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵))
26 rexanali 3146 . . . . . . . . . 10 (∃𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2725, 26bitri 264 . . . . . . . . 9 (∃𝑦 ∈ (𝐴𝐵)Pred(𝑅, (𝐴𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
284, 27sylib 208 . . . . . . . 8 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅)) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))
2928ex 397 . . . . . . 7 ((𝑅 We 𝐴𝑅 Se 𝐴) → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
303, 29mpani 676 . . . . . 6 ((𝑅 We 𝐴𝑅 Se 𝐴) → ((𝐴𝐵) ≠ ∅ → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
312, 30syl5bi 232 . . . . 5 ((𝑅 We 𝐴𝑅 Se 𝐴) → (¬ 𝐴𝐵 → ¬ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)))
3231con4d 115 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → (∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵) → 𝐴𝐵))
3332imp 393 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴𝐵)
3433adantrl 695 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴𝐵)
35 simprl 754 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐵𝐴)
3634, 35eqssd 3769 1 (((𝑅 We 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 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  cdif 3720  cin 3722  wss 3723  c0 4063  {csn 4316   Se wse 5206   We wwe 5207  ccnv 5248  cima 5252  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823
This theorem is referenced by:  wfii  5857  wfisg  5858
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