Theorem frind 13939

Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 13938). 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 is itself equal to A. Compare wfi 13915 and tfi 3937, which are special cases of this theorem that do not require the transitive closure to prove.

Assertion

Ref	Expression
frind	|- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ (B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B))) -> A = B)

Distinct variable groups:   x,A   y,A   y,B   x,R   y,R

Proof of Theorem frind

Step	Hyp	Ref	Expression
1		difss 2735	. . . . . . 7 |- (A \ B) C_ A
2		frmin 13938	. . . . . . . . 9 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ ((A \ B) C_ A /\ (A \ B) =/= (/))) -> E.y e. (A \ B)Pred(R, (A \ B), y) = (/))
3		eldif 2609	. . . . . . . . . . . . 13 |- (y e. (A \ B) <-> (y e. A /\ -. y e. B))
4	3	anbi1i 539	. . . . . . . . . . . 12 |- ((y e. (A \ B) /\ Pred(R, (A \ B), y) = (/)) <-> ((y e. A /\ -. y e. B) /\ Pred(R, (A \ B), y) = (/)))
5		anass 487	. . . . . . . . . . . 12 |- (((y e. A /\ -. y e. B) /\ Pred(R, (A \ B), y) = (/)) <-> (y e. A /\ (-. y e. B /\ Pred(R, (A \ B), y) = (/))))
6		ancom 482	. . . . . . . . . . . . . 14 |- ((-. y e. B /\ Pred(R, (A \ B), y) = (/)) <-> (Pred(R, (A \ B), y) = (/) /\ -. y e. B))
7		indif2 10150	. . . . . . . . . . . . . . . . . 18 |- ((`'R"{y}) i^i (A \ B)) = (((`'R"{y}) i^i A) \ B)
8		df-pred 13880	. . . . . . . . . . . . . . . . . . 19 |- Pred(R, (A \ B), y) = ((A \ B) i^i (`'R"{y}))
9		incom 2787	. . . . . . . . . . . . . . . . . . 19 |- ((A \ B) i^i (`'R"{y})) = ((`'R"{y}) i^i (A \ B))
10	8, 9	eqtri 1908	. . . . . . . . . . . . . . . . . 18 |- Pred(R, (A \ B), y) = ((`'R"{y}) i^i (A \ B))
11		df-pred 13880	. . . . . . . . . . . . . . . . . . . 20 |- Pred(R, A, y) = (A i^i (`'R"{y}))
12		incom 2787	. . . . . . . . . . . . . . . . . . . 20 |- (A i^i (`'R"{y})) = ((`'R"{y}) i^i A)
13	11, 12	eqtri 1908	. . . . . . . . . . . . . . . . . . 19 |- Pred(R, A, y) = ((`'R"{y}) i^i A)
14	13	difeq1i 2722	. . . . . . . . . . . . . . . . . 18 |- (Pred(R, A, y) \ B) = (((`'R"{y}) i^i A) \ B)
15	7, 10, 14	3eqtr4i 1921	. . . . . . . . . . . . . . . . 17 |- Pred(R, (A \ B), y) = (Pred(R, A, y) \ B)
16	15	eqeq1i 1891	. . . . . . . . . . . . . . . 16 |- (Pred(R, (A \ B), y) = (/) <-> (Pred(R, A, y) \ B) = (/))
17		ssdif0 2934	. . . . . . . . . . . . . . . 16 |- (Pred(R, A, y) C_ B <-> (Pred(R, A, y) \ B) = (/))
18	16, 17	bitr4i 193	. . . . . . . . . . . . . . 15 |- (Pred(R, (A \ B), y) = (/) <-> Pred(R, A, y) C_ B)
19	18	anbi1i 539	. . . . . . . . . . . . . 14 |- ((Pred(R, (A \ B), y) = (/) /\ -. y e. B) <-> (Pred(R, A, y) C_ B /\ -. y e. B))
20	6, 19	bitri 190	. . . . . . . . . . . . 13 |- ((-. y e. B /\ Pred(R, (A \ B), y) = (/)) <-> (Pred(R, A, y) C_ B /\ -. y e. B))
21	20	anbi2i 538	. . . . . . . . . . . 12 |- ((y e. A /\ (-. y e. B /\ Pred(R, (A \ B), y) = (/))) <-> (y e. A /\ (Pred(R, A, y) C_ B /\ -. y e. B)))
22	4, 5, 21	3bitri 194	. . . . . . . . . . 11 |- ((y e. (A \ B) /\ Pred(R, (A \ B), y) = (/)) <-> (y e. A /\ (Pred(R, A, y) C_ B /\ -. y e. B)))
23	22	rexbii2 2132	. . . . . . . . . 10 |- (E.y e. (A \ B)Pred(R, (A \ B), y) = (/) <-> E.y e. A (Pred(R, A, y) C_ B /\ -. y e. B))
24		rexanali 2144	. . . . . . . . . 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))
25	23, 24	bitri 190	. . . . . . . . 9 |- (E.y e. (A \ B)Pred(R, (A \ B), y) = (/) <-> -. A.y e. A (Pred(R, A, y) C_ B -> y e. B))
26	2, 25	sylib 215	. . . . . . . 8 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ ((A \ B) C_ A /\ (A \ B) =/= (/))) -> -. A.y e. A (Pred(R, A, y) C_ B -> y e. B))
27	26	ex 402	. . . . . . 7 |- ((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) -> (((A \ B) C_ A /\ (A \ B) =/= (/)) -> -. A.y e. A (Pred(R, A, y) C_ B -> y e. B)))
28	1, 27	mpani 762	. . . . . 6 |- ((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) -> ((A \ B) =/= (/) -> -. A.y e. A (Pred(R, A, y) C_ B -> y e. B)))
29		ssdif0 2934	. . . . . . 7 |- (A C_ B <-> (A \ B) = (/))
30	29	necon3bbii 2031	. . . . . 6 |- (-. A C_ B <-> (A \ B) =/= (/))
31	28, 30	syl5ib 223	. . . . 5 |- ((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) -> (-. A C_ B -> -. A.y e. A (Pred(R, A, y) C_ B -> y e. B)))
32	31	con4d 91	. . . 4 |- ((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) -> (A.y e. A (Pred(R, A, y) C_ B -> y e. B) -> A C_ B))
33	32	imp 377	. . 3 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B)) -> A C_ B)
34	33	adantrl 430	. 2 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ (B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B))) -> A C_ B)
35		simprl 450	. 2 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ (B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B))) -> B C_ A)
36	34, 35	eqssd 2633	1 |- (((R Fr A /\ A.x e. A Pred(R, A, x) e. _V) /\ (B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B))) -> A = B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   Fr wfr 3623  `'ccnv 3985  "cima 3989  Predcpred 13879

This theorem is referenced by:  frindi 13940  frinsg 13941

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-15 1751  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-pred 13880  df-trcl 13925

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