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Theorem wfi 6352
Description: The Principle of Well-Ordered 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.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
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 wefr 5667 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 482 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5668 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5608 . . . . 5 (𝑅 Or 𝐴𝑅 Po 𝐴)
53, 4syl 17 . . . 4 (𝑅 We 𝐴𝑅 Po 𝐴)
65adantr 482 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7 simpr 486 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
82, 6, 73jca 1129 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → (𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴))
9 frpoind 6344 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
108, 9sylan 581 1 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wss 3949   Po wpo 5587   Or wor 5588   Fr wfr 5629   Se wse 5630   We wwe 5631  Predcpred 6300
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301
This theorem is referenced by:  wfii  6354  wfisgOLD  6356
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