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Theorem uniwun 9441
Description: Every set is contained in a weak universe. This is the analogue of grothtsk 9536 for weak universes, but it is provable in ZFC without the Tarski-Grothendieck axiom, contrary to grothtsk 9536. (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 3178 . 2 ( WUni = V ↔ ∀𝑥 𝑥 WUni)
2 snex 4835 . . . 4 {𝑥} ∈ V
3 wunex 9440 . . . 4 ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
42, 3ax-mp 5 . . 3 𝑢 ∈ WUni {𝑥} ⊆ 𝑢
5 eluni2 4376 . . . 4 (𝑥 WUni ↔ ∃𝑢 ∈ WUni 𝑥𝑢)
6 vex 3176 . . . . . 6 𝑥 ∈ V
76snss 4259 . . . . 5 (𝑥𝑢 ↔ {𝑥} ⊆ 𝑢)
87rexbii 3023 . . . 4 (∃𝑢 ∈ WUni 𝑥𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
95, 8bitri 263 . . 3 (𝑥 WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
104, 9mpbir 220 . 2 𝑥 WUni
111, 10mpgbir 1717 1 WUni = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  wrex 2897  Vcvv 3173  wss 3540  {csn 4125   cuni 4372  WUnicwun 9401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-wun 9403
This theorem is referenced by: (None)
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