Theorem tskr1om2 8931

Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7840.) (Contributed by NM, 22-Feb-2011.)

Assertion

Ref	Expression
tskr1om2	|- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Proof of Theorem tskr1om2

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

Step	Hyp	Ref	Expression
1		eluni2 4092	. . 3 |- ( y e. U. ( R1 " om ) <-> E. x e. ( R1 " om ) y e. x )
2		r1fnon 7970	. . . . . . . . 9 |- R1 Fn On
3		fnfun 5505	. . . . . . . . 9 |- ( R1 Fn On -> Fun R1 )
4	2, 3	ax-mp 5	. . . . . . . 8 |- Fun R1
5		fvelima 5740	. . . . . . . 8 |- ( ( Fun R1 /\ x e. ( R1 " om ) ) -> E. y e. om ( R1 ` y ) = x )
6	4, 5	mpan 665	. . . . . . 7 |- ( x e. ( R1 " om ) -> E. y e. om ( R1 ` y ) = x )
7		r1tr 7979	. . . . . . . . 9 |- Tr ( R1 ` y )
8		treq 4388	. . . . . . . . 9 |- ( ( R1 ` y ) = x -> ( Tr ( R1 ` y ) <-> Tr x ) )
9	7, 8	mpbii 211	. . . . . . . 8 |- ( ( R1 ` y ) = x -> Tr x )
10	9	rexlimivw 2835	. . . . . . 7 |- ( E. y e. om ( R1 ` y ) = x -> Tr x )
11		trss 4391	. . . . . . 7 |- ( Tr x -> ( y e. x -> y C_ x ) )
12	6, 10, 11	3syl 20	. . . . . 6 |- ( x e. ( R1 " om ) -> ( y e. x -> y C_ x ) )
13	12	adantl 463	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y C_ x ) )
14		tskr1om 8930	. . . . . . . 8 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( R1 " om ) C_ T )
15	14	sseld 3352	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> x e. T ) )
16		tskss 8921	. . . . . . . . 9 |- ( ( T e. Tarski /\ x e. T /\ y C_ x ) -> y e. T )
17	16	3exp 1181	. . . . . . . 8 |- ( T e. Tarski -> ( x e. T -> ( y C_ x -> y e. T ) ) )
18	17	adantr 462	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. T -> ( y C_ x -> y e. T ) ) )
19	15, 18	syld 44	. . . . . 6 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> ( y C_ x -> y e. T ) ) )
20	19	imp 429	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y C_ x -> y e. T ) )
21	13, 20	syld 44	. . . 4 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y e. T ) )
22	21	rexlimdva 2839	. . 3 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( E. x e. ( R1 " om ) y e. x -> y e. T ) )
23	1, 22	syl5bi 217	. 2 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( y e. U. ( R1 " om ) -> y e. T ) )
24	23	ssrdv 3359	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   E.wrex 2714   C_ wss 3325   (/)c0 3634   U.cuni 4088   Tr wtr 4382   Oncon0 4715   "cima 4839   Fun wfun 5409   Fn wfn 5410   ` cfv 5415   omcom 6475   R1cr1 7965   Tarskictsk 8911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967  df-tsk 8912

This theorem is referenced by: (None)