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Theorem frindi 24315
Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 24313). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is itself equal to  A. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
frind.1  |-  R  Fr  A
frind.2  |-  R Se  A
Assertion
Ref Expression
frindi  |-  ( ( 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 frindi
StepHypRef Expression
1 frind.1 . 2  |-  R  Fr  A
2 frind.2 . 2  |-  R Se  A
3 frind 24314 . 2  |-  ( ( ( R  Fr  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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    Fr wfr 4365   Se wse 4366   Predcpred 24238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-pred 24239  df-trpred 24292
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