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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 9967 | . . . . . . . 8 ⊢ ∅ ∈ Tarski | |
2 | eleq1 2847 | . . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 ∈ Tarski ↔ ∅ ∈ Tarski)) | |
3 | 1, 2 | mpbiri 250 | . . . . . . 7 ⊢ (𝑦 = ∅ → 𝑦 ∈ Tarski) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 = ∅ → 𝑦 ∈ Tarski)) |
5 | vex 3412 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
6 | unir1 9028 | . . . . . . . . . . 11 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2859 | . . . . . . . . . 10 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On) |
8 | eqid 2772 | . . . . . . . . . . 11 ⊢ (𝑦 ∩ On) = (𝑦 ∩ On) | |
9 | 8 | grur1 10032 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
10 | 7, 9 | mpan2 678 | . . . . . . . . 9 ⊢ (𝑦 ∈ Univ → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
11 | 10 | adantr 473 | . . . . . . . 8 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 = (𝑅1‘(𝑦 ∩ On))) |
12 | 8 | gruina 10030 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → (𝑦 ∩ On) ∈ Inacc) |
13 | inatsk 9990 | . . . . . . . . 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 2860 | . . . . . . 7 ⊢ ((𝑦 ∈ Univ ∧ 𝑦 ≠ ∅) → 𝑦 ∈ Tarski) |
16 | 15 | ex 405 | . . . . . 6 ⊢ (𝑦 ∈ Univ → (𝑦 ≠ ∅ → 𝑦 ∈ Tarski)) |
17 | 4, 16 | pm2.61dne 3048 | . . . . 5 ⊢ (𝑦 ∈ Univ → 𝑦 ∈ Tarski) |
18 | grutr 10005 | . . . . 5 ⊢ (𝑦 ∈ Univ → Tr 𝑦) | |
19 | 17, 18 | jca 504 | . . . 4 ⊢ (𝑦 ∈ Univ → (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
20 | grutsk1 10033 | . . . 4 ⊢ ((𝑦 ∈ Tarski ∧ Tr 𝑦) → 𝑦 ∈ Univ) | |
21 | 19, 20 | impbii 201 | . . 3 ⊢ (𝑦 ∈ Univ ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
22 | treq 5030 | . . . 4 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
23 | 22 | elrab 3589 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥} ↔ (𝑦 ∈ Tarski ∧ Tr 𝑦)) |
24 | 21, 23 | bitr4i 270 | . 2 ⊢ (𝑦 ∈ Univ ↔ 𝑦 ∈ {𝑥 ∈ Tarski ∣ Tr 𝑥}) |
25 | 24 | eqriv 2769 | 1 ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 {crab 3086 Vcvv 3409 ∩ cin 3824 ∅c0 4173 ∪ cuni 4706 Tr wtr 5024 “ cima 5403 Oncon0 6023 ‘cfv 6182 𝑅1cr1 8977 Inacccina 9895 Tarskictsk 9960 Univcgru 10002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-reg 8843 ax-inf2 8890 ax-ac2 9675 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-smo 7780 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-oi 8761 df-har 8809 df-tc 8965 df-r1 8979 df-rank 8980 df-card 9154 df-aleph 9155 df-cf 9156 df-acn 9157 df-ac 9328 df-wina 9896 df-ina 9897 df-tsk 9961 df-gru 10003 |
This theorem is referenced by: (None) |
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