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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 10334 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
2 | eleq1 2818 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
3 | 1, 2 | mpbiri 261 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) |
5 | vex 3402 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
6 | unir1 9394 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2830 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) |
8 | eqid 2736 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
9 | 8 | grur1 10399 | . . . . . . . . . 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 484 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
12 | 8 | gruina 10397 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) |
13 | inatsk 10357 | . . . . . . . . 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 2831 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) |
16 | 15 | ex 416 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) |
17 | 4, 16 | pm2.61dne 3018 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) |
18 | grutr 10372 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
19 | 17, 18 | jca 515 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
20 | grutsk1 10400 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
21 | 19, 20 | impbii 212 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
22 | treq 5152 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
23 | 22 | elrab 3591 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
24 | 21, 23 | bitr4i 281 | . 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 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 {crab 3055 Vcvv 3398 ∩ cin 3852 ∅c0 4223 ∪ cuni 4805 Tr wtr 5146 “ cima 5539 Oncon0 6191 ‘cfv 6358 𝑅1cr1 9343 Inacccina 10262 Tarskictsk 10327 Univcgru 10369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 ax-ac2 10042 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-smo 8061 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-oi 9104 df-har 9151 df-tc 9331 df-r1 9345 df-rank 9346 df-card 9520 df-aleph 9521 df-cf 9522 df-acn 9523 df-ac 9695 df-wina 10263 df-ina 10264 df-tsk 10328 df-gru 10370 |
This theorem is referenced by: (None) |
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