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Theorem wfii 24208
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 24207 . 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 663 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 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   Se wse 4350    We wwe 4351   Predcpred 24167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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