Theorem wunr1om 10613

Description: A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypothesis

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)

Assertion

Ref	Expression
wunr1om	⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Proof of Theorem wunr1om

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

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈))
5		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7		r10 9664	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		wun0.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni)
9	8	wun0 10612	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈)
10	7, 9	eqeltrid 2832	. . . . . 6 ⊢ (𝜑 → (𝑅1‘∅) ∈ 𝑈)
11	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘𝑦) ∈ 𝑈)
13	11, 12	wunpw 10601	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈)
14		nnon 7805	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
15		r1suc 9666	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
17	16	eleq1d 2813	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑈))
18	13, 17	imbitrrid 246	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
19	18	expd 415	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑅1‘𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
20	2, 4, 6, 10, 19	finds2 7831	. . . . 5 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈))
21		eleq1 2816	. . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
22	21	imbi2d 340	. . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝜑 → (𝑅1‘𝑥) ∈ 𝑈) ↔ (𝜑 → 𝑦 ∈ 𝑈)))
23	20, 22	syl5ibcom 245	. . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)))
24	23	rexlimiv 3123	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈))
25		r1fnon 9663	. . . . 5 ⊢ 𝑅1 Fn On
26		fnfun 6582	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
27	25, 26	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
28		fvelima 6888	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
29	27, 28	mpan 690	. . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
30	24, 29	syl11 33	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑈))
31	30	ssrdv 3941	1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551   “ cima 5622  Oncon0 6307  suc csuc 6309  Fun wfun 6476   Fn wfn 6477  ‘cfv 6482  ωcom 7799  𝑅₁cr1 9658  WUnicwun 10594

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-r1 9660  df-wun 10596

This theorem is referenced by:  wunom  10614