Theorem tskr1om 10720

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 9591.) (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 6858	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘𝑦) ∈ 𝑇))
5		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7		r10 9721	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		tsk0 10716	. . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
9	7, 8	eqeltrid 2832	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10		tskpw 10706	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → 𝒫 (𝑅1‘𝑦) ∈ 𝑇)
11		nnon 7848	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
12		r1suc 9723	. . . . . . . . . . 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 2813	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑇))
15	10, 14	imbitrrid 246	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇))
16	15	expd 415	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
17	16	adantrd 491	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇)))
18	2, 4, 6, 9, 17	finds2 7874	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇))
19		eleq1 2816	. . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
20	19	imbi2d 340	. . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
21	18, 20	syl5ibcom 245	. . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)))
22	21	rexlimiv 3127	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))
23		r1fnon 9720	. . . . 5 ⊢ 𝑅1 Fn On
24		fnfun 6618	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
25	23, 24	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
26		fvelima 6926	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
27	25, 26	mpan 690	. . 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 3952	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563   “ cima 5641  Oncon0 6332  suc csuc 6334  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  ωcom 7842  𝑅₁cr1 9715  Tarskictsk 10701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-tsk 10702

This theorem is referenced by:  tskr1om2  10721  tskinf  10722