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Theorem grutsk1 10250
 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 10212.) (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 10182 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
32adantlr 714 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
4 tskpr 10199 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
543expa 1115 . . . . . 6 (((𝑇 ∈ Tarski ∧ 𝑥𝑇) ∧ 𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
65ralrimiva 3149 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
76adantlr 714 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
8 elmapg 8420 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
98adantlr 714 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
10 tskurn 10218 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇𝑦:𝑥𝑇) → ran 𝑦𝑇)
11103expia 1118 . . . . . 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 3148 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇)
143, 7, 133jca 1125 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
1514ralrimiva 3149 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
16 elgrug 10221 . . 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 712 1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  𝒫 cpw 4500  {cpr 4530  ∪ cuni 4804  Tr wtr 5140  ran crn 5524  ⟶wf 6328  (class class class)co 7145   ↑m cmap 8407  Tarskictsk 10177  Univcgru 10219 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-ac2 9892 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-smo 7984  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-oi 8976  df-har 9023  df-r1 9195  df-card 9370  df-aleph 9371  df-cf 9372  df-acn 9373  df-ac 9545  df-wina 10113  df-ina 10114  df-tsk 10178  df-gru 10220 This theorem is referenced by:  grutsk  10251  inagrud  41175
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