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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 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑈 ∈ WUni) |
3 | 1 | wunr1om 10663 | . . . . . 6 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
4 | r1funlim 9710 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpli 485 | . . . . . . 7 ⊢ Fun 𝑅1 |
6 | 4 | simpri 487 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
7 | limomss 7811 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
9 | funimass4 6911 | . . . . . . 7 ⊢ ((Fun 𝑅1 ∧ ω ⊆ dom 𝑅1) → ((𝑅1 “ ω) ⊆ 𝑈 ↔ ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈)) | |
10 | 5, 8, 9 | mp2an 691 | . . . . . 6 ⊢ ((𝑅1 “ ω) ⊆ 𝑈 ↔ ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈) |
11 | 3, 10 | sylib 217 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈) |
12 | 11 | r19.21bi 3233 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑅1‘𝑥) ∈ 𝑈) |
13 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ ω) | |
14 | 8, 13 | sselid 3946 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ dom 𝑅1) |
15 | onssr1 9775 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ⊆ (𝑅1‘𝑥)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ⊆ (𝑅1‘𝑥)) |
17 | 2, 12, 16 | wunss 10656 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ 𝑈) |
18 | 17 | ex 414 | . 2 ⊢ (𝜑 → (𝑥 ∈ ω → 𝑥 ∈ 𝑈)) |
19 | 18 | ssrdv 3954 | 1 ⊢ (𝜑 → ω ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3061 ⊆ wss 3914 dom cdm 5637 “ cima 5640 Lim wlim 6322 Fun wfun 6494 ‘cfv 6500 ωcom 7806 𝑅1cr1 9706 WUnicwun 10644 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-r1 9708 df-rank 9709 df-wun 10646 |
This theorem is referenced by: (None) |
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