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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 9456 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
2 | eleq1 2676 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
3 | 1, 2 | mpbiri 247 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) |
5 | vex 3176 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
6 | unir1 8559 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2687 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) |
8 | eqid 2610 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
9 | 8 | grur1 9521 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
10 | 7, 9 | mpan2 703 | . . . . . . . . 9 ⊢ (𝑦 ∈ Univ → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
11 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
12 | 8 | gruina 9519 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) |
13 | inatsk 9479 | . . . . . . . . 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 2688 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) |
16 | 15 | ex 449 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) |
17 | 4, 16 | pm2.61dne 2868 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) |
18 | grutr 9494 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
19 | 17, 18 | jca 553 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
20 | grutsk1 9522 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
21 | 19, 20 | impbii 198 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
22 | treq 4686 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
23 | 22 | elrab 3331 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
24 | 21, 23 | bitr4i 266 | . 2 ⊢ (𝑦 ∈ Univ ↔ 𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥}) |
25 | 24 | eqriv 2607 | 1 ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∩ cin 3539 ∅c0 3874 ∪ cuni 4372 Tr wtr 4680 “ cima 5041 Oncon0 5640 ‘cfv 5804 𝑅1cr1 8508 Inacccina 9384 Tarskictsk 9449 Univcgru 9491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-smo 7330 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-tc 8496 df-r1 8510 df-rank 8511 df-card 8648 df-aleph 8649 df-cf 8650 df-acn 8651 df-ac 8822 df-wina 9385 df-ina 9386 df-tsk 9450 df-gru 9492 |
This theorem is referenced by: (None) |
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