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Theorem grutsk1 9522
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 9484.) (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 476 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → Tr 𝑇)
2 tskpw 9454 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
32adantlr 747 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
4 tskpr 9471 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
543expa 1257 . . . . . 6 (((𝑇 ∈ Tarski ∧ 𝑥𝑇) ∧ 𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
65ralrimiva 2949 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
76adantlr 747 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
8 elmapg 7757 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) ↔ 𝑦:𝑥𝑇))
98adantlr 747 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) ↔ 𝑦:𝑥𝑇))
10 tskurn 9490 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇𝑦:𝑥𝑇) → ran 𝑦𝑇)
11103expia 1259 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦:𝑥𝑇 ran 𝑦𝑇))
129, 11sylbid 229 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) → ran 𝑦𝑇))
1312ralrimiv 2948 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇)
143, 7, 133jca 1235 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))
1514ralrimiva 2949 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))
16 elgrug 9493 . . 3 (𝑇 ∈ Tarski → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))))
1716adantr 480 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))))
181, 15, 17mpbir2and 959 1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wcel 1977  wral 2896  𝒫 cpw 4108  {cpr 4127   cuni 4372  Tr wtr 4680  ran crn 5039  wf 5800  (class class class)co 6549  𝑚 cmap 7744  Tarskictsk 9449  Univcgru 9491
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  ax-ac2 9168
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-rmo 2904  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-int 4411  df-iun 4457  df-iin 4458  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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-r1 8510  df-card 8648  df-aleph 8649  df-cf 8650  df-acn 8651  df-ac 8822  df-wina 9385  df-ina 9386  df-tsk 9450  df-gru 9492
This theorem is referenced by:  grutsk  9523
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