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Theorem wfii 25463
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.)
Hypotheses
Ref Expression
wfi.1  |-  R  We  A
wfi.2  |-  R Se  A
Assertion
Ref Expression
wfii  |-  ( ( B  C_  A  /\  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
Distinct variable groups:    y, A    y, B    y, R

Proof of Theorem wfii
StepHypRef Expression
1 wfi.1 . 2  |-  R  We  A
2 wfi.2 . 2  |-  R Se  A
3 wfi 25462 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
41, 2, 3mpanl12 664 1  |-  ( ( B  C_  A  /\  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   Se wse 4531    We wwe 4532   Predcpred 25422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-pred 25423
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