Theorem tskr1om2 9590

Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8535.) (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 4440	. . 3 |- ( y e. U. ( R1 " om ) <-> E. x e. ( R1 " om ) y e. x )
2		r1fnon 8630	. . . . . . . . 9 |- R1 Fn On
3		fnfun 5988	. . . . . . . . 9 |- ( R1 Fn On -> Fun R1 )
4	2, 3	ax-mp 5	. . . . . . . 8 |- Fun R1
5		fvelima 6248	. . . . . . . 8 |- ( ( Fun R1 /\ x e. ( R1 " om ) ) -> E. y e. om ( R1 ` y ) = x )
6	4, 5	mpan 706	. . . . . . 7 |- ( x e. ( R1 " om ) -> E. y e. om ( R1 ` y ) = x )
7		r1tr 8639	. . . . . . . . 9 |- Tr ( R1 ` y )
8		treq 4758	. . . . . . . . 9 |- ( ( R1 ` y ) = x -> ( Tr ( R1 ` y ) <-> Tr x ) )
9	7, 8	mpbii 223	. . . . . . . 8 |- ( ( R1 ` y ) = x -> Tr x )
10	9	rexlimivw 3029	. . . . . . 7 |- ( E. y e. om ( R1 ` y ) = x -> Tr x )
11		trss 4761	. . . . . . 7 |- ( Tr x -> ( y e. x -> y C_ x ) )
12	6, 10, 11	3syl 18	. . . . . 6 |- ( x e. ( R1 " om ) -> ( y e. x -> y C_ x ) )
13	12	adantl 482	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y C_ x ) )
14		tskr1om 9589	. . . . . . . 8 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( R1 " om ) C_ T )
15	14	sseld 3602	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> x e. T ) )
16		tskss 9580	. . . . . . . . 9 |- ( ( T e. Tarski /\ x e. T /\ y C_ x ) -> y e. T )
17	16	3exp 1264	. . . . . . . 8 |- ( T e. Tarski -> ( x e. T -> ( y C_ x -> y e. T ) ) )
18	17	adantr 481	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. T -> ( y C_ x -> y e. T ) ) )
19	15, 18	syld 47	. . . . . 6 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> ( y C_ x -> y e. T ) ) )
20	19	imp 445	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y C_ x -> y e. T ) )
21	13, 20	syld 47	. . . 4 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y e. T ) )
22	21	rexlimdva 3031	. . 3 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( E. x e. ( R1 " om ) y e. x -> y e. T ) )
23	1, 22	syl5bi 232	. 2 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( y e. U. ( R1 " om ) -> y e. T ) )
24	23	ssrdv 3609	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   U.cuni 4436   Tr wtr 4752   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   R1cr1 8625   Tarskictsk 9570

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-tsk 9571

This theorem is referenced by: (None)