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Mirrors > Home > MPE Home > Th. List > grutsk | Structured version Visualization version GIF version |
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 10179 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
2 | eleq1 2902 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
3 | 1, 2 | mpbiri 260 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) |
5 | vex 3499 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
6 | unir1 9244 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2914 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) |
8 | eqid 2823 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
9 | 8 | grur1 10244 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
10 | 7, 9 | mpan2 689 | . . . . . . . . 9 ⊢ (𝑦 ∈ Univ → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
11 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
12 | 8 | gruina 10242 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) |
13 | inatsk 10202 | . . . . . . . . 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 2915 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) |
16 | 15 | ex 415 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) |
17 | 4, 16 | pm2.61dne 3105 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) |
18 | grutr 10217 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
19 | 17, 18 | jca 514 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
20 | grutsk1 10245 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
21 | 19, 20 | impbii 211 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
22 | treq 5180 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
23 | 22 | elrab 3682 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
24 | 21, 23 | bitr4i 280 | . 2 ⊢ (𝑦 ∈ Univ ↔ 𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥}) |
25 | 24 | eqriv 2820 | 1 ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 ∩ cin 3937 ∅c0 4293 ∪ cuni 4840 Tr wtr 5174 “ cima 5560 Oncon0 6193 ‘cfv 6357 𝑅1cr1 9193 Inacccina 10107 Tarskictsk 10172 Univcgru 10214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-smo 7985 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-har 9024 df-tc 9181 df-r1 9195 df-rank 9196 df-card 9370 df-aleph 9371 df-cf 9372 df-acn 9373 df-ac 9544 df-wina 10108 df-ina 10109 df-tsk 10173 df-gru 10215 |
This theorem is referenced by: (None) |
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