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Theorem wfii 23563
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 23562 . 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 666 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516    C_ wss 3113   Se wse 4308    We wwe 4309   Predcpred 23522
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-pred 23523
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