Theorem wfis 5716

Description: Well-Founded Induction Schema. If all elements less than a given set x of the well-founded class A have a property (induction hypothesis), then all elements of A have that property. (Contributed by Scott Fenton, 29-Jan-2011.)

Hypotheses

Ref	Expression
wfis.1	|- R We A
wfis.2	|- R Se A
wfis.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )

Assertion

Ref	Expression
wfis	|- ( y e. A -> ph )

Distinct variable groups:   y, A, z   ph, z   y, R, z

Allowed substitution hint:   ph( y)

Proof of Theorem wfis

Step	Hyp	Ref	Expression
1		wfis.1	. . 3 |- R We A
2		wfis.2	. . 3 |- R Se A
3		wfis.3	. . . 4 |- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
4	3	wfisg 5715	. . 3 |- ( ( R We A /\ R Se A ) -> A. y e. A ph )
5	1, 2, 4	mp2an 708	. 2 |- A. y e. A ph
6	5	rspec 2931	1 |- ( y e. A -> ph )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   [.wsbc 3435   Se wse 5071   We wwe 5072   Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by: (None)