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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 10780 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
2 | eleq1 2813 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
3 | 1, 2 | mpbiri 257 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) |
5 | vex 3465 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
6 | unir1 9838 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2824 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) |
8 | eqid 2725 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
9 | 8 | grur1 10845 | . . . . . . . . . 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 479 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
12 | 8 | gruina 10843 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) |
13 | inatsk 10803 | . . . . . . . . 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 2825 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) |
16 | 15 | ex 411 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) |
17 | 4, 16 | pm2.61dne 3017 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) |
18 | grutr 10818 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
19 | 17, 18 | jca 510 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
20 | grutsk1 10846 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
21 | 19, 20 | impbii 208 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
22 | treq 5274 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
23 | 22 | elrab 3679 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
24 | 21, 23 | bitr4i 277 | . 2 ⊢ (𝑦 ∈ Univ ↔ 𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥}) |
25 | 24 | eqriv 2722 | 1 ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {crab 3418 Vcvv 3461 ∩ cin 3943 ∅c0 4322 ∪ cuni 4909 Tr wtr 5266 “ cima 5681 Oncon0 6371 ‘cfv 6549 𝑅1cr1 9787 Inacccina 10708 Tarskictsk 10773 Univcgru 10815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-reg 9617 ax-inf2 9666 ax-ac2 10488 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-smo 8367 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9535 df-har 9582 df-tc 9762 df-r1 9789 df-rank 9790 df-card 9964 df-aleph 9965 df-cf 9966 df-acn 9967 df-ac 10141 df-wina 10709 df-ina 10710 df-tsk 10774 df-gru 10816 |
This theorem is referenced by: (None) |
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