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Mirrors > Home > MPE Home > Th. List > wunom | Structured version Visualization version GIF version |
Description: A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
Ref | Expression |
---|---|
wunom | ⊢ (𝜑 → ω ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | 1 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑈 ∈ WUni) |
3 | 1 | wunr1om 9733 | . . . . . 6 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
4 | r1funlim 8802 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpli 476 | . . . . . . 7 ⊢ Fun 𝑅1 |
6 | 4 | simpri 481 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
7 | limomss 7235 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
9 | funimass4 6409 | . . . . . . 7 ⊢ ((Fun 𝑅1 ∧ ω ⊆ dom 𝑅1) → ((𝑅1 “ ω) ⊆ 𝑈 ↔ ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈)) | |
10 | 5, 8, 9 | mp2an 710 | . . . . . 6 ⊢ ((𝑅1 “ ω) ⊆ 𝑈 ↔ ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈) |
11 | 3, 10 | sylib 208 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈) |
12 | 11 | r19.21bi 3070 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑅1‘𝑥) ∈ 𝑈) |
13 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ ω) | |
14 | 8, 13 | sseldi 3742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ dom 𝑅1) |
15 | onssr1 8867 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ⊆ (𝑅1‘𝑥)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ⊆ (𝑅1‘𝑥)) |
17 | 2, 12, 16 | wunss 9726 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ 𝑈) |
18 | 17 | ex 449 | . 2 ⊢ (𝜑 → (𝑥 ∈ ω → 𝑥 ∈ 𝑈)) |
19 | 18 | ssrdv 3750 | 1 ⊢ (𝜑 → ω ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2139 ∀wral 3050 ⊆ wss 3715 dom cdm 5266 “ cima 5269 Lim wlim 5885 Fun wfun 6043 ‘cfv 6049 ωcom 7230 𝑅1cr1 8798 WUnicwun 9714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-r1 8800 df-rank 8801 df-wun 9716 |
This theorem is referenced by: (None) |
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