Theorem wunr1om 10628

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 6832	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3		fveq2 6832	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈))
5		fveq2 6832	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7		r10 9678	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		wun0.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni)
9	8	wun0 10627	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈)
10	7, 9	eqeltrid 2838	. . . . . 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 10616	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈)
14		nnon 7812	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
15		r1suc 9680	. . . . . . . . . 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 2819	. . . . . . . 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 7838	. . . . 5 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈))
21		eleq1 2822	. . . . . 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 3128	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈))
25		r1fnon 9677	. . . . 5 ⊢ 𝑅1 Fn On
26		fnfun 6590	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
27	25, 26	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
28		fvelima 6897	. . . 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 3937	1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552   “ cima 5625  Oncon0 6315  suc csuc 6317  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490  ωcom 7806  𝑅₁cr1 9672  WUnicwun 10609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9674  df-wun 10611

This theorem is referenced by:  wunom  10629