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| Description: Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| grutsk | ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0tsk 10796 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
| 2 | eleq1 2828 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
| 3 | 1, 2 | mpbiri 258 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) | 
| 4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) | 
| 5 | vex 3483 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
| 6 | unir1 9854 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
| 7 | 5, 6 | eleqtrri 2839 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) | 
| 8 | eqid 2736 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
| 9 | 8 | grur1 10861 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) | 
| 10 | 7, 9 | mpan2 691 | . . . . . . . . 9 ⊢ (𝑦 ∈ Univ → 𝑦 = (𝑅1‘(𝑦 ∩ On))) | 
| 11 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) | 
| 12 | 8 | gruina 10859 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) | 
| 13 | inatsk 10819 | . . . . . . . . 9 ⊢ ((𝑦 ∩ On) ∈ Inacc → (𝑅1‘(𝑦 ∩ On)) ∈ Tarski) | |
| 14 | 12, 13 | syl 17 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑅1‘(𝑦 ∩ On)) ∈ Tarski) | 
| 15 | 11, 14 | eqeltrd 2840 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) | 
| 16 | 15 | ex 412 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) | 
| 17 | 4, 16 | pm2.61dne 3027 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) | 
| 18 | grutr 10834 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
| 19 | 17, 18 | jca 511 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) | 
| 20 | grutsk1 10862 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
| 21 | 19, 20 | impbii 209 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) | 
| 22 | treq 5266 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
| 23 | 22 | elrab 3691 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) | 
| 24 | 21, 23 | bitr4i 278 | . 2 ⊢ (𝑦 ∈ Univ ↔ 𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥}) | 
| 25 | 24 | eqriv 2733 | 1 ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {crab 3435 Vcvv 3479 ∩ cin 3949 ∅c0 4332 ∪ cuni 4906 Tr wtr 5258 “ cima 5687 Oncon0 6383 ‘cfv 6560 𝑅1cr1 9803 Inacccina 10724 Tarskictsk 10789 Univcgru 10831 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 ax-ac2 10504 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-smo 8387 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-oi 9551 df-har 9598 df-tc 9778 df-r1 9805 df-rank 9806 df-card 9980 df-aleph 9981 df-cf 9982 df-acn 9983 df-ac 10157 df-wina 10725 df-ina 10726 df-tsk 10790 df-gru 10832 | 
| This theorem is referenced by: (None) | 
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