Theorem r1tr 8265

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 8255	. . . . . 6 |- ( Fun R1 /\ Lim dom R1 )
2	1	simpri 469	. . . . 5 |- Lim dom R1
3		limord 5489	. . . . 5 |- ( Lim dom R1 -> Ord dom R1 )
4		ordsson 6635	. . . . 5 |- ( Ord dom R1 -> dom R1 C_ On )
5	2, 3, 4	mp2b 10	. . . 4 |- dom R1 C_ On
6	5	sseli 3414	. . 3 |- ( A e. dom R1 -> A e. On )
7		fveq2 5879	. . . . . 6 |- ( x = (/) -> ( R1 ` x ) = ( R1 ` (/) ) )
8		r10 8257	. . . . . 6 |- ( R1 ` (/) ) = (/)
9	7, 8	syl6eq 2521	. . . . 5 |- ( x = (/) -> ( R1 ` x ) = (/) )
10		treq 4496	. . . . 5 |- ( ( R1 ` x ) = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
11	9, 10	syl 17	. . . 4 |- ( x = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
12		fveq2 5879	. . . . 5 |- ( x = y -> ( R1 ` x ) = ( R1 ` y ) )
13		treq 4496	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
14	12, 13	syl 17	. . . 4 |- ( x = y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
15		fveq2 5879	. . . . 5 |- ( x = suc y -> ( R1 ` x ) = ( R1 ` suc y ) )
16		treq 4496	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` suc y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
17	15, 16	syl 17	. . . 4 |- ( x = suc y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
18		fveq2 5879	. . . . 5 |- ( x = A -> ( R1 ` x ) = ( R1 ` A ) )
19		treq 4496	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` A ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
20	18, 19	syl 17	. . . 4 |- ( x = A -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
21		tr0 4501	. . . 4 |- Tr (/)
22		limsuc 6695	. . . . . . . 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 468	. . . . . . . . 9 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` y ) )
25		pwtr 4653	. . . . . . . . 9 |- ( Tr ( R1 ` y ) <-> Tr ~P ( R1 ` y ) )
26	24, 25	sylib 201	. . . . . . . 8 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ~P ( R1 ` y ) )
27		r1sucg 8258	. . . . . . . . 9 |- ( y e. dom R1 -> ( R1 ` suc y ) = ~P ( R1 ` y ) )
28		treq 4496	. . . . . . . . 9 |- ( ( R1 ` suc y ) = ~P ( R1 ` y ) -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
29	27, 28	syl 17	. . . . . . . 8 |- ( y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
30	26, 29	syl5ibrcom 230	. . . . . . 7 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( y e. dom R1 -> Tr ( R1 ` suc y ) ) )
31	23, 30	syl5bir 226	. . . . . 6 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( suc y e. dom R1 -> Tr ( R1 ` suc y ) ) )
32		ndmfv 5903	. . . . . . . 8 |- ( -. suc y e. dom R1 -> ( R1 ` suc y ) = (/) )
33		treq 4496	. . . . . . . 8 |- ( ( R1 ` suc y ) = (/) -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
34	32, 33	syl 17	. . . . . . 7 |- ( -. suc y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
35	21, 34	mpbiri 241	. . . . . 6 |- ( -. suc y e. dom R1 -> Tr ( R1 ` suc y ) )
36	31, 35	pm2.61d1 164	. . . . 5 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` suc y ) )
37	36	ex 441	. . . 4 |- ( y e. On -> ( Tr ( R1 ` y ) -> Tr ( R1 ` suc y ) ) )
38		triun 4503	. . . . . . . 8 |- ( A. y e. x Tr ( R1 ` y ) -> Tr U_ y e. x ( R1 ` y ) )
39		r1limg 8260	. . . . . . . . . 10 |- ( ( x e. dom R1 /\ Lim x ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
40	39	ancoms 460	. . . . . . . . 9 |- ( ( Lim x /\ x e. dom R1 ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
41		treq 4496	. . . . . . . . 9 |- ( ( R1 ` x ) = U_ y e. x ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
42	40, 41	syl 17	. . . . . . . 8 |- ( ( Lim x /\ x e. dom R1 ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
43	38, 42	syl5ibr 229	. . . . . . 7 |- ( ( Lim x /\ x e. dom R1 ) -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
44	43	impancom 447	. . . . . 6 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> ( x e. dom R1 -> Tr ( R1 ` x ) ) )
45		ndmfv 5903	. . . . . . . 8 |- ( -. x e. dom R1 -> ( R1 ` x ) = (/) )
46	45, 10	syl 17	. . . . . . 7 |- ( -. x e. dom R1 -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
47	21, 46	mpbiri 241	. . . . . 6 |- ( -. x e. dom R1 -> Tr ( R1 ` x ) )
48	44, 47	pm2.61d1 164	. . . . 5 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> Tr ( R1 ` x ) )
49	48	ex 441	. . . 4 |- ( Lim x -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
50	11, 14, 17, 20, 21, 37, 49	tfinds 6705	. . 3 |- ( A e. On -> Tr ( R1 ` A ) )
51	6, 50	syl 17	. 2 |- ( A e. dom R1 -> Tr ( R1 ` A ) )
52		ndmfv 5903	. . . 4 |- ( -. A e. dom R1 -> ( R1 ` A ) = (/) )
53		treq 4496	. . . 4 |- ( ( R1 ` A ) = (/) -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
54	52, 53	syl 17	. . 3 |- ( -. A e. dom R1 -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
55	21, 54	mpbiri 241	. 2 |- ( -. A e. dom R1 -> Tr ( R1 ` A ) )
56	51, 55	pm2.61i 169	1 |- Tr ( R1 ` A )

Syntax hints:   -. wn 3   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U_ciun 4269   Tr wtr 4490   dom cdm 4839   Ord word 5429   Oncon0 5430   Lim wlim 5431   suc csuc 5432   Fun wfun 5583   ` cfv 5589   R1cr1 8251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-r1 8253

This theorem is referenced by:  r1tr2  8266  r1ordg  8267  r1ord3g  8268  r1ord2  8270  r1sssuc  8272  r1pwss  8273  r1val1  8275  rankwflemb  8282  r1elwf  8285  r1elssi  8294  uniwf  8308  tcrank  8373  ackbij2lem3  8689  r1limwun  9179  tskr1om2  9211