Theorem tskr1om 9589

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 8535.) (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 6191	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘𝑦) ∈ 𝑇))
5		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7		r10 8631	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		tsk0 9585	. . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
9	7, 8	syl5eqel 2705	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10		tskpw 9575	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → 𝒫 (𝑅1‘𝑦) ∈ 𝑇)
11		nnon 7071	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
12		r1suc 8633	. . . . . . . . . . 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 2686	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑇))
15	10, 14	syl5ibr 236	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
16	15	expd 452	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
17	16	adantrd 484	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
18	2, 4, 6, 9, 17	finds2 7094	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇))
19		eleq1 2689	. . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
20	19	imbi2d 330	. . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
21	18, 20	syl5ibcom 235	. . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
22	21	rexlimiv 3027	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))
23		r1fnon 8630	. . . . 5 ⊢ 𝑅1 Fn On
24		fnfun 5988	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
25	23, 24	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
26		fvelima 6248	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
27	25, 26	mpan 706	. . 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 3609	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   “ cima 5117  Oncon0 5723  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  ωcom 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:  tskr1om2  9590  tskinf  9591