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Mirrors > Home > MPE Home > Th. List > wunccl | Structured version Visualization version GIF version |
Description: The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wunccl | ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wuncval 10526 | . 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) | |
2 | ssrab2 4016 | . . 3 ⊢ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ⊆ WUni | |
3 | wunex 10523 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | |
4 | rabn0 4322 | . . . 4 ⊢ ({𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅ ↔ ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | |
5 | 3, 4 | sylibr 233 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅) |
6 | intwun 10519 | . . 3 ⊢ (({𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ⊆ WUni ∧ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅) → ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ∈ WUni) | |
7 | 2, 5, 6 | sylancr 586 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ∈ WUni) |
8 | 1, 7 | eqeltrd 2834 | 1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2101 ≠ wne 2938 ∃wrex 3068 {crab 3221 ⊆ wss 3889 ∅c0 4259 ∩ cint 4882 ‘cfv 6447 WUnicwun 10484 wUniClcwunm 10485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-om 7733 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-wun 10486 df-wunc 10487 |
This theorem is referenced by: wuncidm 10530 wuncval2 10531 |
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