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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 | fveq2 6884 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2812 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈)) |
3 | fveq2 6884 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦)) | |
4 | 3 | eleq1d 2812 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈)) |
5 | fveq2 6884 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦)) | |
6 | 5 | eleq1d 2812 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈)) |
7 | r10 9762 | . . . . . . 7 ⊢ (𝑅1‘∅) = ∅ | |
8 | wun0.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
9 | 8 | wun0 10712 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈) |
10 | 7, 9 | eqeltrid 2831 | . . . . . 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 10701 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈) |
14 | nnon 7857 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On) | |
15 | r1suc 9764 | . . . . . . . . . 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 2812 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑈)) |
18 | 13, 17 | imbitrrid 245 | . . . . . . 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 7887 | . . . . 5 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈)) |
21 | eleq1 2815 | . . . . . 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 244 | . . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈))) |
24 | 23 | rexlimiv 3142 | . . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)) |
25 | r1fnon 9761 | . . . . 5 ⊢ 𝑅1 Fn On | |
26 | fnfun 6642 | . . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ Fun 𝑅1 |
28 | fvelima 6950 | . . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) | |
29 | 27, 28 | mpan 687 | . . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) |
30 | 24, 29 | syl11 33 | . 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑈)) |
31 | 30 | ssrdv 3983 | 1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 “ cima 5672 Oncon0 6357 suc csuc 6359 Fun wfun 6530 Fn wfn 6531 ‘cfv 6536 ωcom 7851 𝑅1cr1 9756 WUnicwun 10694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-r1 9758 df-wun 10696 |
This theorem is referenced by: wunom 10714 |
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