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Theorem grutsk1 10859
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 10821.) (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 484 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → Tr 𝑇)
2 tskpw 10791 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
32adantlr 715 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
4 tskpr 10808 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
543expa 1117 . . . . . 6 (((𝑇 ∈ Tarski ∧ 𝑥𝑇) ∧ 𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
65ralrimiva 3144 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
76adantlr 715 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
8 elmapg 8878 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
98adantlr 715 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) ↔ 𝑦:𝑥𝑇))
10 tskurn 10827 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇𝑦:𝑥𝑇) → ran 𝑦𝑇)
11103expia 1120 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦:𝑥𝑇 ran 𝑦𝑇))
129, 11sylbid 240 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇m 𝑥) → ran 𝑦𝑇))
1312ralrimiv 3143 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇)
143, 7, 133jca 1127 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
1514ralrimiva 3144 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇m 𝑥) ran 𝑦𝑇))
16 elgrug 10830 . . 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 713 1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wcel 2106  wral 3059  𝒫 cpw 4605  {cpr 4633   cuni 4912  Tr wtr 5265  ran crn 5690  wf 6559  (class class class)co 7431  m cmap 8865  Tarskictsk 10786  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-smo 8385  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-r1 9802  df-card 9977  df-aleph 9978  df-cf 9979  df-acn 9980  df-ac 10154  df-wina 10722  df-ina 10723  df-tsk 10787  df-gru 10829
This theorem is referenced by:  grutsk  10860  inagrud  44292
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