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Mirrors > Home > MPE Home > Th. List > uniwun | Structured version Visualization version GIF version |
Description: Every set is contained in a weak universe. This is the analogue of grothtsk 10246 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10246. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
uniwun | ⊢ ∪ WUni = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3449 | . 2 ⊢ (∪ WUni = V ↔ ∀𝑥 𝑥 ∈ ∪ WUni) | |
2 | snex 5297 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | wunex 10150 | . . . 4 ⊢ ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢 |
5 | eluni2 4804 | . . . 4 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni 𝑥 ∈ 𝑢) | |
6 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | snss 4679 | . . . . 5 ⊢ (𝑥 ∈ 𝑢 ↔ {𝑥} ⊆ 𝑢) |
8 | 7 | rexbii 3210 | . . . 4 ⊢ (∃𝑢 ∈ WUni 𝑥 ∈ 𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
9 | 5, 8 | bitri 278 | . . 3 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
10 | 4, 9 | mpbir 234 | . 2 ⊢ 𝑥 ∈ ∪ WUni |
11 | 1, 10 | mpgbir 1801 | 1 ⊢ ∪ WUni = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 {csn 4525 ∪ cuni 4800 WUnicwun 10111 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-wun 10113 |
This theorem is referenced by: (None) |
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