Theorem tskr1om2 9144

Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8053.) (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 4234	. . 3 |- ( y e. U. ( R1 " om ) <-> E. x e. ( R1 " om ) y e. x )
2		r1fnon 8183	. . . . . . . . 9 |- R1 Fn On
3		fnfun 5664	. . . . . . . . 9 |- ( R1 Fn On -> Fun R1 )
4	2, 3	ax-mp 5	. . . . . . . 8 |- Fun R1
5		fvelima 5906	. . . . . . . 8 |- ( ( Fun R1 /\ x e. ( R1 " om ) ) -> E. y e. om ( R1 ` y ) = x )
6	4, 5	mpan 670	. . . . . . 7 |- ( x e. ( R1 " om ) -> E. y e. om ( R1 ` y ) = x )
7		r1tr 8192	. . . . . . . . 9 |- Tr ( R1 ` y )
8		treq 4532	. . . . . . . . 9 |- ( ( R1 ` y ) = x -> ( Tr ( R1 ` y ) <-> Tr x ) )
9	7, 8	mpbii 211	. . . . . . . 8 |- ( ( R1 ` y ) = x -> Tr x )
10	9	rexlimivw 2930	. . . . . . 7 |- ( E. y e. om ( R1 ` y ) = x -> Tr x )
11		trss 4535	. . . . . . 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 466	. . . . 5 |- ( ( ( T e. Tarski /\ T =/= (/) ) /\ x e. ( R1 " om ) ) -> ( y e. x -> y C_ x ) )
14		tskr1om 9143	. . . . . . . 8 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( R1 " om ) C_ T )
15	14	sseld 3485	. . . . . . 7 |- ( ( T e. Tarski /\ T =/= (/) ) -> ( x e. ( R1 " om ) -> x e. T ) )
16		tskss 9134	. . . . . . . . 9 |- ( ( T e. Tarski /\ x e. T /\ y C_ x ) -> y e. T )
17	16	3exp 1194	. . . . . . . 8 |- ( T e. Tarski -> ( x e. T -> ( y C_ x -> y e. T ) ) )
18	17	adantr 465	. . . . . . 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 2933	. . 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 3492	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> U. ( R1 " om ) C_ T )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   =/= wne 2636   E.wrex 2792   C_ wss 3458   (/)c0 3767   U.cuni 4230   Tr wtr 4526   Oncon0 4864   "cima 4988   Fun wfun 5568   Fn wfn 5569   ` cfv 5574   omcom 6681   R1cr1 8178   Tarskictsk 9124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180  df-tsk 9125

This theorem is referenced by: (None)