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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 6843 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2823 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈)) |
3 | fveq2 6843 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦)) | |
4 | 3 | eleq1d 2823 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈)) |
5 | fveq2 6843 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦)) | |
6 | 5 | eleq1d 2823 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈)) |
7 | r10 9705 | . . . . . . 7 ⊢ (𝑅1‘∅) = ∅ | |
8 | wun0.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
9 | 8 | wun0 10655 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈) |
10 | 7, 9 | eqeltrid 2842 | . . . . . 6 ⊢ (𝜑 → (𝑅1‘∅) ∈ 𝑈) |
11 | 8 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝑈 ∈ WUni) |
12 | simpr 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘𝑦) ∈ 𝑈) | |
13 | 11, 12 | wunpw 10644 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈) |
14 | nnon 7809 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On) | |
15 | r1suc 9707 | . . . . . . . . . 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 2823 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑈)) |
18 | 13, 17 | syl5ibr 246 | . . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈)) |
19 | 18 | expd 417 | . . . . . 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 2826 | . . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ 𝑦 ∈ 𝑈)) | |
22 | 21 | imbi2d 341 | . . . . 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 3146 | . . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)) |
25 | r1fnon 9704 | . . . . 5 ⊢ 𝑅1 Fn On | |
26 | fnfun 6603 | . . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ Fun 𝑅1 |
28 | fvelima 6909 | . . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) | |
29 | 27, 28 | mpan 689 | . . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) |
30 | 24, 29 | syl11 33 | . 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑈)) |
31 | 30 | ssrdv 3951 | 1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ⊆ wss 3911 ∅c0 4283 𝒫 cpw 4561 “ cima 5637 Oncon0 6318 suc csuc 6320 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 ωcom 7803 𝑅1cr1 9699 WUnicwun 10637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-r1 9701 df-wun 10639 |
This theorem is referenced by: wunom 10657 |
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