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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gruex | Structured version Visualization version GIF version |
Description: Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
gruex | ⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankon 9208 | . . 3 ⊢ (rank‘𝑥) ∈ On | |
2 | inaex 41005 | . . 3 ⊢ ((rank‘𝑥) ∈ On → ∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧 |
4 | simplr 768 | . . . . . 6 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝑧) | |
5 | inawina 10101 | . . . . . . . . 9 ⊢ (𝑧 ∈ Inacc → 𝑧 ∈ Inaccw) | |
6 | winaon 10099 | . . . . . . . . 9 ⊢ (𝑧 ∈ Inaccw → 𝑧 ∈ On) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ Inacc → 𝑧 ∈ On) |
8 | 7 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑧 ∈ On) |
9 | vex 3444 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
10 | 9 | rankr1a 9249 | . . . . . . 7 ⊢ (𝑧 ∈ On → (𝑥 ∈ (𝑅1‘𝑧) ↔ (rank‘𝑥) ∈ 𝑧)) |
11 | 8, 10 | syl 17 | . . . . . 6 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → (𝑥 ∈ (𝑅1‘𝑧) ↔ (rank‘𝑥) ∈ 𝑧)) |
12 | 4, 11 | mpbird 260 | . . . . 5 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑥 ∈ (𝑅1‘𝑧)) |
13 | simpr 488 | . . . . 5 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑦 = (𝑅1‘𝑧)) | |
14 | 12, 13 | eleqtrrd 2893 | . . . 4 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑥 ∈ 𝑦) |
15 | simpl 486 | . . . . 5 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → 𝑧 ∈ Inacc) | |
16 | 15 | inagrud 41004 | . . . 4 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → (𝑅1‘𝑧) ∈ Univ) |
17 | 14, 16 | rspcime 3575 | . . 3 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦) |
18 | 17 | rexlimiva 3240 | . 2 ⊢ (∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧 → ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦) |
19 | 3, 18 | ax-mp 5 | 1 ⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Oncon0 6159 ‘cfv 6324 𝑅1cr1 9175 rankcrnk 9176 Inaccwcwina 10093 Inacccina 10094 Univcgru 10201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 ax-groth 10234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-smo 7966 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-har 9005 df-r1 9177 df-rank 9178 df-card 9352 df-aleph 9353 df-cf 9354 df-acn 9355 df-ac 9527 df-wina 10095 df-ina 10096 df-tsk 10160 df-gru 10202 |
This theorem is referenced by: rr-groth 41007 rr-grothprim 41008 |
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