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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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑈 ∈ WUni) |
3 | 1 | wunr1om 10654 | . . . . . 6 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
4 | r1funlim 9701 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpli 484 | . . . . . . 7 ⊢ Fun 𝑅1 |
6 | 4 | simpri 486 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
7 | limomss 7806 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
9 | funimass4 6907 | . . . . . . 7 ⊢ ((Fun 𝑅1 ∧ ω ⊆ dom 𝑅1) → ((𝑅1 “ ω) ⊆ 𝑈 ↔ ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈)) | |
10 | 5, 8, 9 | mp2an 690 | . . . . . 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 3234 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑅1‘𝑥) ∈ 𝑈) |
13 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ ω) | |
14 | 8, 13 | sselid 3942 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ dom 𝑅1) |
15 | onssr1 9766 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ⊆ (𝑅1‘𝑥)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ⊆ (𝑅1‘𝑥)) |
17 | 2, 12, 16 | wunss 10647 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ 𝑈) |
18 | 17 | ex 413 | . 2 ⊢ (𝜑 → (𝑥 ∈ ω → 𝑥 ∈ 𝑈)) |
19 | 18 | ssrdv 3950 | 1 ⊢ (𝜑 → ω ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3910 dom cdm 5633 “ cima 5636 Lim wlim 6318 Fun wfun 6490 ‘cfv 6496 ωcom 7801 𝑅1cr1 9697 WUnicwun 10635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-r1 9699 df-rank 9700 df-wun 10637 |
This theorem is referenced by: (None) |
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