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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 10779 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10779. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
uniwun | ⊢ ∪ WUni = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3456 | . 2 ⊢ (∪ WUni = V ↔ ∀𝑥 𝑥 ∈ ∪ WUni) | |
2 | vsnex 5390 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | wunex 10683 | . . . 4 ⊢ ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢 |
5 | eluni2 4873 | . . . 4 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni 𝑥 ∈ 𝑢) | |
6 | vex 3451 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | snss 4750 | . . . . 5 ⊢ (𝑥 ∈ 𝑢 ↔ {𝑥} ⊆ 𝑢) |
8 | 7 | rexbii 3094 | . . . 4 ⊢ (∃𝑢 ∈ WUni 𝑥 ∈ 𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
9 | 5, 8 | bitri 275 | . . 3 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
10 | 4, 9 | mpbir 230 | . 2 ⊢ 𝑥 ∈ ∪ WUni |
11 | 1, 10 | mpgbir 1802 | 1 ⊢ ∪ WUni = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∃wrex 3070 Vcvv 3447 ⊆ wss 3914 {csn 4590 ∪ cuni 4869 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 ax-inf2 9585 |
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-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-1o 8416 df-wun 10646 |
This theorem is referenced by: (None) |
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