Theorem wfi 5713

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.)

Assertion

Ref	Expression
wfi	|- ( ( ( R We A /\ R Se A ) /\ ( 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 wfi

Step	Hyp	Ref	Expression
1		ssdif0 3942	. . . . . . 7 |- ( A C_ B <-> ( A \ B ) = (/) )
2	1	necon3bbii 2841	. . . . . 6 |- ( -. A C_ B <-> ( A \ B ) =/= (/) )
3		difss 3737	. . . . . . 7 |- ( A \ B ) C_ A
4		tz6.26 5711	. . . . . . . . 9 |- ( ( ( R We A /\ R Se A ) /\ ( ( A \ B ) C_ A /\ ( A \ B ) =/= (/) ) ) -> E. y e. ( A \ B ) Pred ( R , ( A \ B ) , y ) = (/) )
5		eldif 3584	. . . . . . . . . . . . 13 |- ( y e. ( A \ B ) <-> ( y e. A /\ -. y e. B ) )
6	5	anbi1i 731	. . . . . . . . . . . 12 |- ( ( y e. ( A \ B ) /\ Pred ( R , ( A \ B ) , y ) = (/) ) <-> ( ( y e. A /\ -. y e. B ) /\ Pred ( R , ( A \ B ) , y ) = (/) ) )
7		anass 681	. . . . . . . . . . . 12 |- ( ( ( y e. A /\ -. y e. B ) /\ Pred ( R , ( A \ B ) , y ) = (/) ) <-> ( y e. A /\ ( -. y e. B /\ Pred ( R , ( A \ B ) , y ) = (/) ) ) )
8		ancom 466	. . . . . . . . . . . . . 14 |- ( ( -. y e. B /\ Pred ( R , ( A \ B ) , y ) = (/) ) <-> ( Pred ( R , ( A \ B ) , y ) = (/) /\ -. y e. B ) )
9		indif2 3870	. . . . . . . . . . . . . . . . . 18 |- ( ( `' R " { y } ) i^i ( A \ B ) ) = ( ( ( `' R " { y } ) i^i A ) \ B )
10		df-pred 5680	. . . . . . . . . . . . . . . . . . 19 |- Pred ( R , ( A \ B ) , y ) = ( ( A \ B ) i^i ( `' R " { y } ) )
11		incom 3805	. . . . . . . . . . . . . . . . . . 19 |- ( ( A \ B ) i^i ( `' R " { y } ) ) = ( ( `' R " { y } ) i^i ( A \ B ) )
12	10, 11	eqtri 2644	. . . . . . . . . . . . . . . . . 18 |- Pred ( R , ( A \ B ) , y ) = ( ( `' R " { y } ) i^i ( A \ B ) )
13		df-pred 5680	. . . . . . . . . . . . . . . . . . . 20 |- Pred ( R , A , y ) = ( A i^i ( `' R " { y } ) )
14		incom 3805	. . . . . . . . . . . . . . . . . . . 20 |- ( A i^i ( `' R " { y } ) ) = ( ( `' R " { y } ) i^i A )
15	13, 14	eqtri 2644	. . . . . . . . . . . . . . . . . . 19 |- Pred ( R , A , y ) = ( ( `' R " { y } ) i^i A )
16	15	difeq1i 3724	. . . . . . . . . . . . . . . . . 18 |- ( Pred ( R , A , y ) \ B ) = ( ( ( `' R " { y } ) i^i A ) \ B )
17	9, 12, 16	3eqtr4i 2654	. . . . . . . . . . . . . . . . 17 |- Pred ( R , ( A \ B ) , y ) = ( Pred ( R , A , y ) \ B )
18	17	eqeq1i 2627	. . . . . . . . . . . . . . . 16 |- ( Pred ( R , ( A \ B ) , y ) = (/) <-> ( Pred ( R , A , y ) \ B ) = (/) )
19		ssdif0 3942	. . . . . . . . . . . . . . . 16 |- ( Pred ( R , A , y ) C_ B <-> ( Pred ( R , A , y ) \ B ) = (/) )
20	18, 19	bitr4i 267	. . . . . . . . . . . . . . 15 |- ( Pred ( R , ( A \ B ) , y ) = (/) <-> Pred ( R , A , y ) C_ B )
21	20	anbi1i 731	. . . . . . . . . . . . . 14 |- ( ( Pred ( R , ( A \ B ) , y ) = (/) /\ -. y e. B ) <-> ( Pred ( R , A , y ) C_ B /\ -. y e. B ) )
22	8, 21	bitri 264	. . . . . . . . . . . . 13 |- ( ( -. y e. B /\ Pred ( R , ( A \ B ) , y ) = (/) ) <-> ( Pred ( R , A , y ) C_ B /\ -. y e. B ) )
23	22	anbi2i 730	. . . . . . . . . . . 12 |- ( ( y e. A /\ ( -. y e. B /\ Pred ( R , ( A \ B ) , y ) = (/) ) ) <-> ( y e. A /\ ( Pred ( R , A , y ) C_ B /\ -. y e. B ) ) )
24	6, 7, 23	3bitri 286	. . . . . . . . . . 11 |- ( ( y e. ( A \ B ) /\ Pred ( R , ( A \ B ) , y ) = (/) ) <-> ( y e. A /\ ( Pred ( R , A , y ) C_ B /\ -. y e. B ) ) )
25	24	rexbii2 3039	. . . . . . . . . 10 |- ( E. y e. ( A \ B ) Pred ( R , ( A \ B ) , y ) = (/) <-> E. y e. A ( Pred ( R , A , y ) C_ B /\ -. y e. B ) )
26		rexanali 2998	. . . . . . . . . 10 |- ( E. y e. A ( Pred ( R , A , y ) C_ B /\ -. y e. B ) <-> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) )
27	25, 26	bitri 264	. . . . . . . . 9 |- ( E. y e. ( A \ B ) Pred ( R , ( A \ B ) , y ) = (/) <-> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) )
28	4, 27	sylib 208	. . . . . . . 8 |- ( ( ( R We A /\ R Se A ) /\ ( ( A \ B ) C_ A /\ ( A \ B ) =/= (/) ) ) -> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) )
29	28	ex 450	. . . . . . 7 |- ( ( R We A /\ R Se A ) -> ( ( ( A \ B ) C_ A /\ ( A \ B ) =/= (/) ) -> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) )
30	3, 29	mpani 712	. . . . . 6 |- ( ( R We A /\ R Se A ) -> ( ( A \ B ) =/= (/) -> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) )
31	2, 30	syl5bi 232	. . . . 5 |- ( ( R We A /\ R Se A ) -> ( -. A C_ B -> -. A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) )
32	31	con4d 114	. . . 4 |- ( ( R We A /\ R Se A ) -> ( A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) -> A C_ B ) )
33	32	imp 445	. . 3 |- ( ( ( R We A /\ R Se A ) /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) -> A C_ B )
34	33	adantrl 752	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A C_ B )
35		simprl 794	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> B C_ A )
36	34, 35	eqssd 3620	1 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   Se wse 5071   We wwe 5072   `'ccnv 5113   "cima 5117   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:  wfii  5714  wfisg  5715