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Theorem intgru 9951
 Description: The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
intgru ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)

Proof of Theorem intgru
Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 479 . . 3 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2 intex 5042 . . 3 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
31, 2sylib 210 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
4 dfss3 3816 . . . . 5 (𝐴 ⊆ Univ ↔ ∀𝑢𝐴 𝑢 ∈ Univ)
5 grutr 9930 . . . . . 6 (𝑢 ∈ Univ → Tr 𝑢)
65ralimi 3161 . . . . 5 (∀𝑢𝐴 𝑢 ∈ Univ → ∀𝑢𝐴 Tr 𝑢)
74, 6sylbi 209 . . . 4 (𝐴 ⊆ Univ → ∀𝑢𝐴 Tr 𝑢)
8 trint 4991 . . . 4 (∀𝑢𝐴 Tr 𝑢 → Tr 𝐴)
97, 8syl 17 . . 3 (𝐴 ⊆ Univ → Tr 𝐴)
109adantr 474 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr 𝐴)
11 grupw 9932 . . . . . . . . . 10 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → 𝒫 𝑥𝑢)
1211ex 403 . . . . . . . . 9 (𝑢 ∈ Univ → (𝑥𝑢 → 𝒫 𝑥𝑢))
1312ral2imi 3156 . . . . . . . 8 (∀𝑢𝐴 𝑢 ∈ Univ → (∀𝑢𝐴 𝑥𝑢 → ∀𝑢𝐴 𝒫 𝑥𝑢))
14 vex 3417 . . . . . . . . 9 𝑥 ∈ V
1514elint2 4704 . . . . . . . 8 (𝑥 𝐴 ↔ ∀𝑢𝐴 𝑥𝑢)
16 vpwex 5077 . . . . . . . . 9 𝒫 𝑥 ∈ V
1716elint2 4704 . . . . . . . 8 (𝒫 𝑥 𝐴 ↔ ∀𝑢𝐴 𝒫 𝑥𝑢)
1813, 15, 173imtr4g 288 . . . . . . 7 (∀𝑢𝐴 𝑢 ∈ Univ → (𝑥 𝐴 → 𝒫 𝑥 𝐴))
1918imp 397 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
2019adantlr 708 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
21 r19.26 3274 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) ↔ (∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢))
22 grupr 9934 . . . . . . . . . . . 12 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23223expia 1156 . . . . . . . . . . 11 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦𝑢 → {𝑥, 𝑦} ∈ 𝑢))
2423ral2imi 3156 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
2521, 24sylbir 227 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
26 vex 3417 . . . . . . . . . 10 𝑦 ∈ V
2726elint2 4704 . . . . . . . . 9 (𝑦 𝐴 ↔ ∀𝑢𝐴 𝑦𝑢)
28 prex 5130 . . . . . . . . . 10 {𝑥, 𝑦} ∈ V
2928elint2 4704 . . . . . . . . 9 ({𝑥, 𝑦} ∈ 𝐴 ↔ ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢)
3025, 27, 293imtr4g 288 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3115, 30sylan2b 589 . . . . . . 7 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3231ralrimiv 3174 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
3332adantlr 708 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
34 elmapg 8135 . . . . . . . . . 10 (( 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3534elvd 3419 . . . . . . . . 9 ( 𝐴 ∈ V → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
362, 35sylbi 209 . . . . . . . 8 (𝐴 ≠ ∅ → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3736ad2antlr 720 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
38 intss1 4712 . . . . . . . . . . . 12 (𝑢𝐴 𝐴𝑢)
39 fss 6291 . . . . . . . . . . . 12 ((𝑦:𝑥 𝐴 𝐴𝑢) → 𝑦:𝑥𝑢)
4038, 39sylan2 588 . . . . . . . . . . 11 ((𝑦:𝑥 𝐴𝑢𝐴) → 𝑦:𝑥𝑢)
4140ralrimiva 3175 . . . . . . . . . 10 (𝑦:𝑥 𝐴 → ∀𝑢𝐴 𝑦:𝑥𝑢)
42 gruurn 9935 . . . . . . . . . . . . . 14 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦:𝑥𝑢) → ran 𝑦𝑢)
43423expia 1156 . . . . . . . . . . . . 13 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦:𝑥𝑢 ran 𝑦𝑢))
4443ral2imi 3156 . . . . . . . . . . . 12 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4521, 44sylbir 227 . . . . . . . . . . 11 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4615, 45sylan2b 589 . . . . . . . . . 10 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4741, 46syl5 34 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 → ∀𝑢𝐴 ran 𝑦𝑢))
4826rnex 7362 . . . . . . . . . . 11 ran 𝑦 ∈ V
4948uniex 7213 . . . . . . . . . 10 ran 𝑦 ∈ V
5049elint2 4704 . . . . . . . . 9 ( ran 𝑦 𝐴 ↔ ∀𝑢𝐴 ran 𝑦𝑢)
5147, 50syl6ibr 244 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5251adantlr 708 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5337, 52sylbid 232 . . . . . 6 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) → ran 𝑦 𝐴))
5453ralrimiv 3174 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴)
5520, 33, 543jca 1164 . . . 4 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
5655ralrimiva 3175 . . 3 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
574, 56sylanb 578 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
58 elgrug 9929 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ Univ ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))))
5958biimpar 471 . 2 (( 𝐴 ∈ V ∧ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))) → 𝐴 ∈ Univ)
603, 10, 57, 59syl12anc 872 1 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   ∈ wcel 2166   ≠ wne 2999  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  𝒫 cpw 4378  {cpr 4399  ∪ cuni 4658  ∩ cint 4697  Tr wtr 4975  ran crn 5343  ⟶wf 6119  (class class class)co 6905   ↑𝑚 cmap 8122  Univcgru 9927 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iin 4743  df-br 4874  df-opab 4936  df-tr 4976  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-gru 9928 This theorem is referenced by: (None)
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