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Theorem uniwun 10154
Description: Every set is contained in a weak universe. This is the analogue of grothtsk 10249 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10249. (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 3507 . 2 ( WUni = V ↔ ∀𝑥 𝑥 WUni)
2 snex 5327 . . . 4 {𝑥} ∈ V
3 wunex 10153 . . . 4 ({𝑥} ∈ V → ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
42, 3ax-mp 5 . . 3 𝑢 ∈ WUni {𝑥} ⊆ 𝑢
5 eluni2 4840 . . . 4 (𝑥 WUni ↔ ∃𝑢 ∈ WUni 𝑥𝑢)
6 vex 3502 . . . . . 6 𝑥 ∈ V
76snss 4716 . . . . 5 (𝑥𝑢 ↔ {𝑥} ⊆ 𝑢)
87rexbii 3251 . . . 4 (∃𝑢 ∈ WUni 𝑥𝑢 ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
95, 8bitri 276 . . 3 (𝑥 WUni ↔ ∃𝑢 ∈ WUni {𝑥} ⊆ 𝑢)
104, 9mpbir 232 . 2 𝑥 WUni
111, 10mpgbir 1793 1 WUni = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  wrex 3143  Vcvv 3499  wss 3939  {csn 4563   cuni 4836  WUnicwun 10114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-wun 10116
This theorem is referenced by: (None)
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