Theorem tskr1om2 9135

Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8046.) (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 4239	. . 3 |- ( y e. U. ( R1 " om ) <-> E. x e. ( R1 " om ) y e. x )
2		r1fnon 8176	. . . . . . . . 9 |- R1 Fn On
3		fnfun 5660	. . . . . . . . 9 |- ( R1 Fn On -> Fun R1 )
4	2, 3	ax-mp 5	. . . . . . . 8 |- Fun R1
5		fvelima 5900	. . . . . . . 8 |- ( ( Fun R1 /\ x e. ( R1 " om ) ) -> E. y e. om ( R1 ` y ) = x )
6	4, 5	mpan 668	. . . . . . 7 |- ( x e. ( R1 " om ) -> E. y e. om ( R1 ` y ) = x )
7		r1tr 8185	. . . . . . . . 9 |- Tr ( R1 ` y )
8		treq 4538	. . . . . . . . 9 |- ( ( R1 ` y ) = x -> ( Tr ( R1 ` y ) <-> Tr x ) )
9	7, 8	mpbii 211	. . . . . . . 8 |- ( ( R1 ` y ) = x -> Tr x )
10	9	rexlimivw 2943	. . . . . . 7 |- ( E. y e. om ( R1 ` y ) = x -> Tr x )
11		trss 4541	. . . . . . 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 464	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y C_ x ) )
14		tskr1om 9134	. . . . . . . 8 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( R1 " om ) C_ T )
15	14	sseld 3488	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> x e. T ) )
16		tskss 9125	. . . . . . . . 9 |- ( ( T e. Tarski /\ x e. T /\ y C_ x ) -> y e. T )
17	16	3exp 1193	. . . . . . . 8 |- ( T e. Tarski -> ( x e. T -> ( y C_ x -> y e. T ) ) )
18	17	adantr 463	. . . . . . 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 427	. . . . 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 2946	. . 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 3495	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   C_ wss 3461   (/)c0 3783   U.cuni 4235   Tr wtr 4532   Oncon0 4867   "cima 4991   Fun wfun 5564   Fn wfn 5565   ` cfv 5570   omcom 6673   R1cr1 8171   Tarskictsk 9115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-tsk 9116

This theorem is referenced by: (None)