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Mirrors > Home > MPE Home > Th. List > wunr1om | Structured version Visualization version GIF version |
Description: A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
Ref | Expression |
---|---|
wunr1om | ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1fnon 8513 | . . . . . 6 ⊢ 𝑅1 Fn On | |
2 | fnfun 5902 | . . . . . 6 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Fun 𝑅1 |
4 | fvelima 6158 | . . . . 5 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) | |
5 | 3, 4 | mpan 702 | . . . 4 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) |
6 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅)) | |
7 | 6 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈)) |
8 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦)) | |
9 | 8 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈)) |
10 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦)) | |
11 | 10 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈)) |
12 | r10 8514 | . . . . . . . 8 ⊢ (𝑅1‘∅) = ∅ | |
13 | wun0.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
14 | 13 | wun0 9419 | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑈) |
15 | 12, 14 | syl5eqel 2692 | . . . . . . 7 ⊢ (𝜑 → (𝑅1‘∅) ∈ 𝑈) |
16 | 13 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝑈 ∈ WUni) |
17 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘𝑦) ∈ 𝑈) | |
18 | 16, 17 | wunpw 9408 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈) |
19 | nnon 6963 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On) | |
20 | r1suc 8516 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦)) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦)) |
22 | 21 | eleq1d 2672 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑈)) |
23 | 18, 22 | syl5ibr 235 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈)) |
24 | 23 | expd 451 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑅1‘𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈))) |
25 | 7, 9, 11, 15, 24 | finds2 6986 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈)) |
26 | eleq1 2676 | . . . . . . 7 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ 𝑦 ∈ 𝑈)) | |
27 | 26 | imbi2d 329 | . . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝜑 → (𝑅1‘𝑥) ∈ 𝑈) ↔ (𝜑 → 𝑦 ∈ 𝑈))) |
28 | 25, 27 | syl5ibcom 234 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈))) |
29 | 28 | rexlimiv 3009 | . . . 4 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)) |
30 | 5, 29 | syl 17 | . . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → (𝜑 → 𝑦 ∈ 𝑈)) |
31 | 30 | com12 32 | . 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑈)) |
32 | 31 | ssrdv 3574 | 1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 “ cima 5041 Oncon0 5640 suc csuc 5642 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 ωcom 6957 𝑅1cr1 8508 WUnicwun 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-wun 9403 |
This theorem is referenced by: wunom 9421 |
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