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Theorem uniwun 10780
Description: Every set is contained in a weak universe. This is the analogue of grothtsk 10875 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10875. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
uniwun WUni = V

Proof of Theorem uniwun
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqv 3490 . 2 ( WUni = V ↔ ∀𝑥 𝑥 WUni)
2 vsnex 5434 . . . 4 {𝑥} ∈ V
3 wunex 10779 . . . 4 ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
42, 3ax-mp 5 . . 3 𝑢 ∈ WUni {𝑥} ⊆ 𝑢
5 eluni2 4911 . . . 4 (𝑥 WUni ↔ ∃𝑢 ∈ WUni 𝑥𝑢)
6 vex 3484 . . . . . 6 𝑥 ∈ V
76snss 4785 . . . . 5 (𝑥𝑢 ↔ {𝑥} ⊆ 𝑢)
87rexbii 3094 . . . 4 (∃𝑢 ∈ WUni 𝑥𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
95, 8bitri 275 . . 3 (𝑥 WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
104, 9mpbir 231 . 2 𝑥 WUni
111, 10mpgbir 1799 1 WUni = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  {csn 4626   cuni 4907  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-wun 10742
This theorem is referenced by: (None)
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