Theorem tskr1om2 9211

Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8161.) (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 4194	. . 3 |- ( y e. U. ( R1 " om ) <-> E. x e. ( R1 " om ) y e. x )
2		r1fnon 8256	. . . . . . . . 9 |- R1 Fn On
3		fnfun 5683	. . . . . . . . 9 |- ( R1 Fn On -> Fun R1 )
4	2, 3	ax-mp 5	. . . . . . . 8 |- Fun R1
5		fvelima 5931	. . . . . . . 8 |- ( ( Fun R1 /\ x e. ( R1 " om ) ) -> E. y e. om ( R1 ` y ) = x )
6	4, 5	mpan 684	. . . . . . 7 |- ( x e. ( R1 " om ) -> E. y e. om ( R1 ` y ) = x )
7		r1tr 8265	. . . . . . . . 9 |- Tr ( R1 ` y )
8		treq 4496	. . . . . . . . 9 |- ( ( R1 ` y ) = x -> ( Tr ( R1 ` y ) <-> Tr x ) )
9	7, 8	mpbii 216	. . . . . . . 8 |- ( ( R1 ` y ) = x -> Tr x )
10	9	rexlimivw 2869	. . . . . . 7 |- ( E. y e. om ( R1 ` y ) = x -> Tr x )
11		trss 4499	. . . . . . 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 473	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y C_ x ) )
14		tskr1om 9210	. . . . . . . 8 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( R1 " om ) C_ T )
15	14	sseld 3417	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> x e. T ) )
16		tskss 9201	. . . . . . . . 9 |- ( ( T e. Tarski /\ x e. T /\ y C_ x ) -> y e. T )
17	16	3exp 1230	. . . . . . . 8 |- ( T e. Tarski -> ( x e. T -> ( y C_ x -> y e. T ) ) )
18	17	adantr 472	. . . . . . 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 436	. . . . 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 2871	. . 3 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( E. x e. ( R1 " om ) y e. x -> y e. T ) )
23	1, 22	syl5bi 225	. 2 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( y e. U. ( R1 " om ) -> y e. T ) )
24	23	ssrdv 3424	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   C_ wss 3390   (/)c0 3722   U.cuni 4190   Tr wtr 4490   "cima 4842   Oncon0 5430   Fun wfun 5583   Fn wfn 5584   ` cfv 5589   omcom 6711   R1cr1 8251   Tarskictsk 9191

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-rep 4508  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  df-tsk 9192

This theorem is referenced by: (None)