Theorem wunr1om 10135

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 6665	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
2	1	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3		fveq2 6665	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈))
5		fveq2 6665	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
6	5	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7		r10 9191	. . . . . . 7 ⊢ (𝑅1‘∅) = ∅
8		wun0.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni)
9	8	wun0 10134	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈)
10	7, 9	eqeltrid 2917	. . . . . 6 ⊢ (𝜑 → (𝑅1‘∅) ∈ 𝑈)
11	8	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘𝑦) ∈ 𝑈)
13	11, 12	wunpw 10123	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈)
14		nnon 7580	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
15		r1suc 9193	. . . . . . . . . 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 2897	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑈))
18	13, 17	syl5ibr 248	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
19	18	expd 418	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑅1‘𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
20	2, 4, 6, 10, 19	finds2 7604	. . . . 5 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈))
21		eleq1 2900	. . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
22	21	imbi2d 343	. . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝜑 → (𝑅1‘𝑥) ∈ 𝑈) ↔ (𝜑 → 𝑦 ∈ 𝑈)))
23	20, 22	syl5ibcom 247	. . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)))
24	23	rexlimiv 3280	. . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈))
25		r1fnon 9190	. . . . 5 ⊢ 𝑅1 Fn On
26		fnfun 6448	. . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1)
27	25, 26	ax-mp 5	. . . 4 ⊢ Fun 𝑅1
28		fvelima 6726	. . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
29	27, 28	mpan 688	. . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦)
30	24, 29	syl11 33	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑈))
31	30	ssrdv 3973	1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539   “ cima 5553  Oncon0 6186  suc csuc 6188  Fun wfun 6344   Fn wfn 6345  ‘cfv 6350  ωcom 7574  𝑅₁cr1 9185  WUnicwun 10116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-r1 9187  df-wun 10118

This theorem is referenced by:  wunom  10136