Theorem tskr1om 10792

Description: A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9663.) (Contributed by Mario Carneiro, 24-Jun-2013.)

Assertion

Ref	Expression
tskr1om	⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Proof of Theorem tskr1om

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

Step	Hyp	Ref	Expression
1		fveq2 6896	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2810	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3		fveq2 6896	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2810	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘𝑦) ∈ 𝑇))
5		fveq2 6896	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2810	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7		r10 9793	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		tsk0 10788	. . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
9	7, 8	eqeltrid 2829	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10		tskpw 10778	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → 𝒫 (𝑅1‘𝑦) ∈ 𝑇)
11		nnon 7877	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
12		r1suc 9795	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
14	13	eleq1d 2810	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑇))
15	10, 14	imbitrrid 245	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
16	15	expd 414	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
17	16	adantrd 490	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
18	2, 4, 6, 9, 17	finds2 7906	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇))
19		eleq1 2813	. . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
20	19	imbi2d 339	. . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
21	18, 20	syl5ibcom 244	. . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
22	21	rexlimiv 3137	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))
23		r1fnon 9792	. . . . 5 ⊢ 𝑅1 Fn On
24		fnfun 6655	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
25	23, 24	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
26		fvelima 6963	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
27	25, 26	mpan 688	. . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
28	22, 27	syl11 33	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑇))
29	28	ssrdv 3982	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∃wrex 3059   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604   “ cima 5681  Oncon0 6371  suc csuc 6373  Fun wfun 6543   Fn wfn 6544  ‘cfv 6549  ωcom 7871  𝑅₁cr1 9787  Tarskictsk 10773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-tsk 10774

This theorem is referenced by:  tskr1om2  10793  tskinf  10794