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Theorem grutsk1 10435
Description: Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10397.) (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
grutsk1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)

Proof of Theorem grutsk1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 488 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → Tr 𝑇)
2 tskpw 10367 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
32adantlr 715 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
4 tskpr 10384 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
543expa 1120 . . . . . 6 (((𝑇 ∈ Tarski ∧ 𝑥𝑇) ∧ 𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
65ralrimiva 3105 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
76adantlr 715 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
8 elmapg 8521 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
98adantlr 715 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
10 tskurn 10403 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇𝑦:𝑥𝑇) → ran 𝑦𝑇)
11103expia 1123 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦:𝑥𝑇 ran 𝑦𝑇))
129, 11sylbid 243 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) → ran 𝑦𝑇))
1312ralrimiv 3104 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇)
143, 7, 133jca 1130 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
1514ralrimiva 3105 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
16 elgrug 10406 . . 3 (𝑇 ∈ Tarski → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))))
1716adantr 484 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))))
181, 15, 17mpbir2and 713 1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089  wcel 2110  wral 3061  𝒫 cpw 4513  {cpr 4543   cuni 4819  Tr wtr 5161  ran crn 5552  wf 6376  (class class class)co 7213  m cmap 8508  Tarskictsk 10362  Univcgru 10404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-ac2 10077
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-smo 8083  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-oi 9126  df-har 9173  df-r1 9380  df-card 9555  df-aleph 9556  df-cf 9557  df-acn 9558  df-ac 9730  df-wina 10298  df-ina 10299  df-tsk 10363  df-gru 10405
This theorem is referenced by:  grutsk  10436  inagrud  41587
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