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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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑈 ∈ WUni) |
3 | 1 | wunr1om 10143 | . . . . . 6 ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) |
4 | r1funlim 9197 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpli 486 | . . . . . . 7 ⊢ Fun 𝑅1 |
6 | 4 | simpri 488 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
7 | limomss 7587 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
9 | funimass4 6732 | . . . . . . 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 220 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝑅1‘𝑥) ∈ 𝑈) |
12 | 11 | r19.21bi 3210 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑅1‘𝑥) ∈ 𝑈) |
13 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ ω) | |
14 | 8, 13 | sseldi 3967 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ dom 𝑅1) |
15 | onssr1 9262 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ⊆ (𝑅1‘𝑥)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ⊆ (𝑅1‘𝑥)) |
17 | 2, 12, 16 | wunss 10136 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ 𝑈) |
18 | 17 | ex 415 | . 2 ⊢ (𝜑 → (𝑥 ∈ ω → 𝑥 ∈ 𝑈)) |
19 | 18 | ssrdv 3975 | 1 ⊢ (𝜑 → ω ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 dom cdm 5557 “ cima 5560 Lim wlim 6194 Fun wfun 6351 ‘cfv 6357 ωcom 7582 𝑅1cr1 9193 WUnicwun 10124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-r1 9195 df-rank 9196 df-wun 10126 |
This theorem is referenced by: (None) |
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