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Theorem grutsk1 9849
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 9811.) (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 471 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → Tr 𝑇)
2 tskpw 9781 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
32adantlr 694 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → 𝒫 𝑥𝑇)
4 tskpr 9798 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
543expa 1111 . . . . . 6 (((𝑇 ∈ Tarski ∧ 𝑥𝑇) ∧ 𝑦𝑇) → {𝑥, 𝑦} ∈ 𝑇)
65ralrimiva 3115 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
76adantlr 694 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇)
8 elmapg 8026 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) ↔ 𝑦:𝑥𝑇))
98adantlr 694 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) ↔ 𝑦:𝑥𝑇))
10 tskurn 9817 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇𝑦:𝑥𝑇) → ran 𝑦𝑇)
11103expia 1114 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦:𝑥𝑇 ran 𝑦𝑇))
129, 11sylbid 230 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝑦 ∈ (𝑇𝑚 𝑥) → ran 𝑦𝑇))
1312ralrimiv 3114 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇)
143, 7, 133jca 1122 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥𝑇) → (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))
1514ralrimiva 3115 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))
16 elgrug 9820 . . 3 (𝑇 ∈ Tarski → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))))
1716adantr 466 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → (𝑇 ∈ Univ ↔ (Tr 𝑇 ∧ ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∀𝑦𝑇 {𝑥, 𝑦} ∈ 𝑇 ∧ ∀𝑦 ∈ (𝑇𝑚 𝑥) ran 𝑦𝑇))))
181, 15, 17mpbir2and 692 1 ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wcel 2145  wral 3061  𝒫 cpw 4298  {cpr 4319   cuni 4575  Tr wtr 4887  ran crn 5251  wf 6026  (class class class)co 6796  𝑚 cmap 8013  Tarskictsk 9776  Univcgru 9818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-ac2 9491
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-smo 7600  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8575  df-har 8623  df-r1 8795  df-card 8969  df-aleph 8970  df-cf 8971  df-acn 8972  df-ac 9143  df-wina 9712  df-ina 9713  df-tsk 9777  df-gru 9819
This theorem is referenced by:  grutsk  9850
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