![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10873 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10873. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
uniwun | ⊢ ∪ WUni = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3488 | . 2 ⊢ (∪ WUni = V ↔ ∀𝑥 𝑥 ∈ ∪ WUni) | |
2 | vsnex 5440 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | wunex 10777 | . . . 4 ⊢ ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢 |
5 | eluni2 4916 | . . . 4 ⊢ (𝑥 ∈ ∪ WUni ↔ ∃𝑢 ∈ WUni 𝑥 ∈ 𝑢) | |
6 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | snss 4790 | . . . . 5 ⊢ (𝑥 ∈ 𝑢 ↔ {𝑥} ⊆ 𝑢) |
8 | 7 | rexbii 3092 | . . . 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 231 | . 2 ⊢ 𝑥 ∈ ∪ WUni |
11 | 1, 10 | mpgbir 1796 | 1 ⊢ ∪ WUni = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 WUnicwun 10738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-wun 10740 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |