Theorem tskr1om 10685

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 9554.) (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 6836	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2822	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇))
3		fveq2 6836	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2822	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘𝑦) ∈ 𝑇))
5		fveq2 6836	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2822	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇))
7		r10 9687	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		tsk0 10681	. . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
9	7, 8	eqeltrid 2841	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇)
10		tskpw 10671	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → 𝒫 (𝑅1‘𝑦) ∈ 𝑇)
11		nnon 7818	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
12		r1suc 9689	. . . . . . . . . . 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 2822	. . . . . . . . 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 7844	. . . . 5 ⊢ (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇))
19		eleq1 2825	. . . . . 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 3132	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))
23		r1fnon 9686	. . . . 5 ⊢ 𝑅1 Fn On
24		fnfun 6594	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
25	23, 24	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
26		fvelima 6901	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
27	25, 26	mpan 691	. . 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 3928	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542   “ cima 5629  Oncon0 6319  suc csuc 6321  Fun wfun 6488   Fn wfn 6489  ‘cfv 6494  ωcom 7812  𝑅₁cr1 9681  Tarskictsk 10666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-r1 9683  df-tsk 10667

This theorem is referenced by:  tskr1om2  10686  tskinf  10687