Theorem wsuclem 31773

Description: Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)

Hypotheses

Ref	Expression
wsuclem.1	|- ( ph -> R We A )
wsuclem.2	|- ( ph -> R Se A )
wsuclem.3	|- ( ph -> X e. V )
wsuclem.4	|- ( ph -> E. w e. A X R w )

Assertion

Ref	Expression
wsuclem	|- ( ph -> E. x e. A ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) )

Distinct variable groups:   x, A, y, z, w   ph, x, y   x, R, y, z, w   x, X, y, z, w

Allowed substitution hints:   ph( z, w)   V( x, y, z, w)

Proof of Theorem wsuclem

Step	Hyp	Ref	Expression
1		wsuclem.1	. . 3 |- ( ph -> R We A )
2		wsuclem.2	. . 3 |- ( ph -> R Se A )
3		predss 5687	. . . 4 |- Pred ( `' R , A , X ) C_ A
4	3	a1i 11	. . 3 |- ( ph -> Pred ( `' R , A , X ) C_ A )
5		wsuclem.3	. . . . 5 |- ( ph -> X e. V )
6		dfpred3g 5691	. . . . 5 |- ( X e. V -> Pred ( `' R , A , X ) = { w e. A | w `' R X } )
7	5, 6	syl 17	. . . 4 |- ( ph -> Pred ( `' R , A , X ) = { w e. A | w `' R X } )
8	5	elexd 3214	. . . . 5 |- ( ph -> X e. _V )
9		wsuclem.4	. . . . 5 |- ( ph -> E. w e. A X R w )
10		rabn0 3958	. . . . . . 7 |- ( { w e. A | w `' R X } =/= (/) <-> E. w e. A w `' R X )
11		brcnvg 5303	. . . . . . . . 9 |- ( ( w e. A /\ X e. _V ) -> ( w `' R X <-> X R w ) )
12	11	ancoms 469	. . . . . . . 8 |- ( ( X e. _V /\ w e. A ) -> ( w `' R X <-> X R w ) )
13	12	rexbidva 3049	. . . . . . 7 |- ( X e. _V -> ( E. w e. A w `' R X <-> E. w e. A X R w ) )
14	10, 13	syl5bb 272	. . . . . 6 |- ( X e. _V -> ( { w e. A | w `' R X } =/= (/) <-> E. w e. A X R w ) )
15	14	biimpar 502	. . . . 5 |- ( ( X e. _V /\ E. w e. A X R w ) -> { w e. A | w `' R X } =/= (/) )
16	8, 9, 15	syl2anc 693	. . . 4 |- ( ph -> { w e. A | w `' R X } =/= (/) )
17	7, 16	eqnetrd 2861	. . 3 |- ( ph -> Pred ( `' R , A , X ) =/= (/) )
18		tz6.26 5711	. . 3 |- ( ( ( R We A /\ R Se A ) /\ ( Pred ( `' R , A , X ) C_ A /\ Pred ( `' R , A , X ) =/= (/) ) ) -> E. x e. Pred ( `' R , A , X ) Pred ( R , Pred ( `' R , A , X ) , x ) = (/) )
19	1, 2, 4, 17, 18	syl22anc 1327	. 2 |- ( ph -> E. x e. Pred ( `' R , A , X ) Pred ( R , Pred ( `' R , A , X ) , x ) = (/) )
20		dfpred3g 5691	. . . . 5 |- ( X e. V -> Pred ( `' R , A , X ) = { y e. A | y `' R X } )
21	5, 20	syl 17	. . . 4 |- ( ph -> Pred ( `' R , A , X ) = { y e. A | y `' R X } )
22	21	rexeqdv 3145	. . 3 |- ( ph -> ( E. x e. Pred ( `' R , A , X ) Pred ( R , Pred ( `' R , A , X ) , x ) = (/) <-> E. x e. { y e. A | y `' R X } Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) )
23		breq1 4656	. . . . 5 |- ( y = x -> ( y `' R X <-> x `' R X ) )
24	23	rexrab 3370	. . . 4 |- ( E. x e. { y e. A | y `' R X } Pred ( R , Pred ( `' R , A , X ) , x ) = (/) <-> E. x e. A ( x `' R X /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) )
25		noel 3919	. . . . . . . . . . . 12 |- -. y e. (/)
26		simp2r 1088	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> Pred ( R , Pred ( `' R , A , X ) , x ) = (/) )
27	26	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> ( y e. Pred ( R , Pred ( `' R , A , X ) , x ) <-> y e. (/) ) )
28	25, 27	mtbiri 317	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> -. y e. Pred ( R , Pred ( `' R , A , X ) , x ) )
29		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
30	29	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> x e. _V )
31		simp3 1063	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> y e. Pred ( `' R , A , X ) )
32		elpredg 5694	. . . . . . . . . . . 12 |- ( ( x e. _V /\ y e. Pred ( `' R , A , X ) ) -> ( y e. Pred ( R , Pred ( `' R , A , X ) , x ) <-> y R x ) )
33	30, 31, 32	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> ( y e. Pred ( R , Pred ( `' R , A , X ) , x ) <-> y R x ) )
34	28, 33	mtbid 314	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) /\ y e. Pred ( `' R , A , X ) ) -> -. y R x )
35	34	3expa 1265	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) ) /\ y e. Pred ( `' R , A , X ) ) -> -. y R x )
36	35	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) ) -> A. y e. Pred ( `' R , A , X ) -. y R x )
37	36	expr 643	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( Pred ( R , Pred ( `' R , A , X ) , x ) = (/) -> A. y e. Pred ( `' R , A , X ) -. y R x ) )
38		simp1rl 1126	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> x e. A )
39		simp1rr 1127	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> x `' R X )
40	5	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. A /\ x `' R X ) ) -> X e. V )
41	40	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> X e. V )
42	29	elpred 5693	. . . . . . . . . . . . 13 |- ( X e. V -> ( x e. Pred ( `' R , A , X ) <-> ( x e. A /\ x `' R X ) ) )
43	41, 42	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> ( x e. Pred ( `' R , A , X ) <-> ( x e. A /\ x `' R X ) ) )
44	38, 39, 43	mpbir2and 957	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> x e. Pred ( `' R , A , X ) )
45		simp3 1063	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> x R y )
46		breq1 4656	. . . . . . . . . . . 12 |- ( z = x -> ( z R y <-> x R y ) )
47	46	rspcev 3309	. . . . . . . . . . 11 |- ( ( x e. Pred ( `' R , A , X ) /\ x R y ) -> E. z e. Pred ( `' R , A , X ) z R y )
48	44, 45, 47	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A /\ x R y ) -> E. z e. Pred ( `' R , A , X ) z R y )
49	48	3expia 1267	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ x `' R X ) ) /\ y e. A ) -> ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) )
50	49	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ x `' R X ) ) -> A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) )
51	50	expr 643	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( x `' R X -> A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) )
52	37, 51	anim12d 586	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( Pred ( R , Pred ( `' R , A , X ) , x ) = (/) /\ x `' R X ) -> ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) )
53	52	ancomsd 470	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x `' R X /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) -> ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) )
54	53	reximdva 3017	. . . 4 |- ( ph -> ( E. x e. A ( x `' R X /\ Pred ( R , Pred ( `' R , A , X ) , x ) = (/) ) -> E. x e. A ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) )
55	24, 54	syl5bi 232	. . 3 |- ( ph -> ( E. x e. { y e. A | y `' R X } Pred ( R , Pred ( `' R , A , X ) , x ) = (/) -> E. x e. A ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) )
56	22, 55	sylbid 230	. 2 |- ( ph -> ( E. x e. Pred ( `' R , A , X ) Pred ( R , Pred ( `' R , A , X ) , x ) = (/) -> E. x e. A ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) )
57	19, 56	mpd 15	1 |- ( ph -> E. x e. A ( A. y e. Pred ( `' R , A , X ) -. y R x /\ A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Se wse 5071   We wwe 5072   `'ccnv 5113   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:  wsuccl  31776  wsuclb  31777