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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 10259 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10259. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
uniwun | ⊢ ∪ WUni = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3504 | . 2 ⊢ (∪ WUni = V ↔ ∀𝑥 𝑥 ∈ ∪ WUni) | |
2 | snex 5334 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | wunex 10163 | . . . 4 ⊢ ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢 |
5 | eluni2 4844 | . . . 4 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni 𝑥 ∈ 𝑢) | |
6 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | snss 4720 | . . . . 5 ⊢ (𝑥 ∈ 𝑢 ↔ {𝑥} ⊆ 𝑢) |
8 | 7 | rexbii 3249 | . . . 4 ⊢ (∃𝑢 ∈ WUni 𝑥 ∈ 𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
9 | 5, 8 | bitri 277 | . . 3 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) |
10 | 4, 9 | mpbir 233 | . 2 ⊢ 𝑥 ∈ ∪ WUni |
11 | 1, 10 | mpgbir 1800 | 1 ⊢ ∪ WUni = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 {csn 4569 ∪ cuni 4840 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 ax-inf2 9106 |
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-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-1o 8104 df-wun 10126 |
This theorem is referenced by: (None) |
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