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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 6906 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅)) | |
| 2 | 1 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈)) |
| 3 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦)) | |
| 4 | 3 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘𝑦) ∈ 𝑈)) |
| 5 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦)) | |
| 6 | 5 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈)) |
| 7 | r10 9808 | . . . . . . 7 ⊢ (𝑅1‘∅) = ∅ | |
| 8 | wun0.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 9 | 8 | wun0 10758 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑈) |
| 10 | 7, 9 | eqeltrid 2845 | . . . . . 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 10747 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅1‘𝑦) ∈ 𝑈) → 𝒫 (𝑅1‘𝑦) ∈ 𝑈) |
| 14 | nnon 7893 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On) | |
| 15 | r1suc 9810 | . . . . . . . . . 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 2826 | . . . . . . . 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 7920 | . . . . 5 ⊢ (𝑥 ∈ ω → (𝜑 → (𝑅1‘𝑥) ∈ 𝑈)) |
| 21 | eleq1 2829 | . . . . . 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 3148 | . . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → (𝜑 → 𝑦 ∈ 𝑈)) |
| 25 | r1fnon 9807 | . . . . 5 ⊢ 𝑅1 Fn On | |
| 26 | fnfun 6668 | . . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
| 27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ Fun 𝑅1 |
| 28 | fvelima 6974 | . . . 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 3989 | 1 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 “ cima 5688 Oncon0 6384 suc csuc 6386 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 ωcom 7887 𝑅1cr1 9802 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-wun 10742 |
| This theorem is referenced by: wunom 10760 |
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