Theorem r1tr 8211

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1tr	|- Tr ( R1 ` A )

Proof of Theorem r1tr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1funlim 8201	. . . . . 6 |- ( Fun R1 /\ Lim dom R1 )
2	1	simpri 462	. . . . 5 |- Lim dom R1
3		limord 4946	. . . . 5 |- ( Lim dom R1 -> Ord dom R1 )
4		ordsson 6624	. . . . 5 |- ( Ord dom R1 -> dom R1 C_ On )
5	2, 3, 4	mp2b 10	. . . 4 |- dom R1 C_ On
6	5	sseli 3495	. . 3 |- ( A e. dom R1 -> A e. On )
7		fveq2 5872	. . . . . 6 |- ( x = (/) -> ( R1 ` x ) = ( R1 ` (/) ) )
8		r10 8203	. . . . . 6 |- ( R1 ` (/) ) = (/)
9	7, 8	syl6eq 2514	. . . . 5 |- ( x = (/) -> ( R1 ` x ) = (/) )
10		treq 4556	. . . . 5 |- ( ( R1 ` x ) = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
11	9, 10	syl 16	. . . 4 |- ( x = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
12		fveq2 5872	. . . . 5 |- ( x = y -> ( R1 ` x ) = ( R1 ` y ) )
13		treq 4556	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
14	12, 13	syl 16	. . . 4 |- ( x = y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
15		fveq2 5872	. . . . 5 |- ( x = suc y -> ( R1 ` x ) = ( R1 ` suc y ) )
16		treq 4556	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` suc y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
17	15, 16	syl 16	. . . 4 |- ( x = suc y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
18		fveq2 5872	. . . . 5 |- ( x = A -> ( R1 ` x ) = ( R1 ` A ) )
19		treq 4556	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` A ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
20	18, 19	syl 16	. . . 4 |- ( x = A -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
21		tr0 4561	. . . 4 |- Tr (/)
22		limsuc 6683	. . . . . . . 8 |- ( Lim dom R1 -> ( y e. dom R1 <-> suc y e. dom R1 ) )
23	2, 22	ax-mp 5	. . . . . . 7 |- ( y e. dom R1 <-> suc y e. dom R1 )
24		simpr 461	. . . . . . . . 9 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` y ) )
25		pwtr 4709	. . . . . . . . 9 |- ( Tr ( R1 ` y ) <-> Tr ~P ( R1 ` y ) )
26	24, 25	sylib 196	. . . . . . . 8 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ~P ( R1 ` y ) )
27		r1sucg 8204	. . . . . . . . 9 |- ( y e. dom R1 -> ( R1 ` suc y ) = ~P ( R1 ` y ) )
28		treq 4556	. . . . . . . . 9 |- ( ( R1 ` suc y ) = ~P ( R1 ` y ) -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
29	27, 28	syl 16	. . . . . . . 8 |- ( y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
30	26, 29	syl5ibrcom 222	. . . . . . 7 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( y e. dom R1 -> Tr ( R1 ` suc y ) ) )
31	23, 30	syl5bir 218	. . . . . 6 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( suc y e. dom R1 -> Tr ( R1 ` suc y ) ) )
32		ndmfv 5896	. . . . . . . 8 |- ( -. suc y e. dom R1 -> ( R1 ` suc y ) = (/) )
33		treq 4556	. . . . . . . 8 |- ( ( R1 ` suc y ) = (/) -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
34	32, 33	syl 16	. . . . . . 7 |- ( -. suc y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
35	21, 34	mpbiri 233	. . . . . 6 |- ( -. suc y e. dom R1 -> Tr ( R1 ` suc y ) )
36	31, 35	pm2.61d1 159	. . . . 5 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` suc y ) )
37	36	ex 434	. . . 4 |- ( y e. On -> ( Tr ( R1 ` y ) -> Tr ( R1 ` suc y ) ) )
38		triun 4563	. . . . . . . 8 |- ( A. y e. x Tr ( R1 ` y ) -> Tr U_ y e. x ( R1 ` y ) )
39		r1limg 8206	. . . . . . . . . 10 |- ( ( x e. dom R1 /\ Lim x ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
40	39	ancoms 453	. . . . . . . . 9 |- ( ( Lim x /\ x e. dom R1 ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
41		treq 4556	. . . . . . . . 9 |- ( ( R1 ` x ) = U_ y e. x ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
42	40, 41	syl 16	. . . . . . . 8 |- ( ( Lim x /\ x e. dom R1 ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
43	38, 42	syl5ibr 221	. . . . . . 7 |- ( ( Lim x /\ x e. dom R1 ) -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
44	43	impancom 440	. . . . . 6 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> ( x e. dom R1 -> Tr ( R1 ` x ) ) )
45		ndmfv 5896	. . . . . . . 8 |- ( -. x e. dom R1 -> ( R1 ` x ) = (/) )
46	45, 10	syl 16	. . . . . . 7 |- ( -. x e. dom R1 -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
47	21, 46	mpbiri 233	. . . . . 6 |- ( -. x e. dom R1 -> Tr ( R1 ` x ) )
48	44, 47	pm2.61d1 159	. . . . 5 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> Tr ( R1 ` x ) )
49	48	ex 434	. . . 4 |- ( Lim x -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
50	11, 14, 17, 20, 21, 37, 49	tfinds 6693	. . 3 |- ( A e. On -> Tr ( R1 ` A ) )
51	6, 50	syl 16	. 2 |- ( A e. dom R1 -> Tr ( R1 ` A ) )
52		ndmfv 5896	. . . 4 |- ( -. A e. dom R1 -> ( R1 ` A ) = (/) )
53		treq 4556	. . . 4 |- ( ( R1 ` A ) = (/) -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
54	52, 53	syl 16	. . 3 |- ( -. A e. dom R1 -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
55	21, 54	mpbiri 233	. 2 |- ( -. A e. dom R1 -> Tr ( R1 ` A ) )
56	51, 55	pm2.61i 164	1 |- Tr ( R1 ` A )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U_ciun 4332   Tr wtr 4550   Ord word 4886   Oncon0 4887   Lim wlim 4888   suc csuc 4889   dom cdm 5008   Fun wfun 5588   ` cfv 5594   R1cr1 8197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199

This theorem is referenced by:  r1tr2  8212  r1ordg  8213  r1ord3g  8214  r1ord2  8216  r1sssuc  8218  r1pwss  8219  r1val1  8221  rankwflemb  8228  r1elwf  8231  r1elssi  8240  uniwf  8254  tcrank  8319  ackbij2lem3  8638  r1limwun  9131  tskr1om2  9163