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Mirrors > Home > MPE Home > Th. List > Mathboxes > goaln0 | Structured version Visualization version GIF version |
Description: The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.) |
Ref | Expression |
---|---|
goaln0 | ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-goal 32702 | . . . 4 ⊢ ∀𝑔𝑖𝐴 = 〈2o, 〈𝑖, 𝐴〉〉 | |
2 | 2on0 8096 | . . . . . . . . . . . 12 ⊢ 2o ≠ ∅ | |
3 | 2 | neii 2989 | . . . . . . . . . . 11 ⊢ ¬ 2o = ∅ |
4 | 3 | intnanr 491 | . . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉) |
5 | 2oex 8095 | . . . . . . . . . . 11 ⊢ 2o ∈ V | |
6 | opex 5321 | . . . . . . . . . . 11 ⊢ 〈𝑖, 𝐴〉 ∈ V | |
7 | 5, 6 | opth 5333 | . . . . . . . . . 10 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 ↔ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉)) |
8 | 4, 7 | mtbir 326 | . . . . . . . . 9 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 |
9 | goel 32707 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = 〈∅, 〈𝑘, 𝑗〉〉) | |
10 | 9 | eqeq2d 2809 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉)) |
11 | 8, 10 | mtbiri 330 | . . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
12 | 11 | rgen2 3168 | . . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
13 | ralnex2 3221 | . . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) | |
14 | 12, 13 | mpbi 233 | . . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
15 | 14 | intnan 490 | . . . . 5 ⊢ ¬ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
16 | eqeq1 2802 | . . . . . . 7 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) | |
17 | 16 | 2rexbidv 3259 | . . . . . 6 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
18 | fmla0 32742 | . . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)} | |
19 | 17, 18 | elrab2 3631 | . . . . 5 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) ↔ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
20 | 15, 19 | mtbir 326 | . . . 4 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) |
21 | 1, 20 | eqneltri 2883 | . . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅) |
22 | fveq2 6645 | . . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅)) | |
23 | 22 | eleq2d 2875 | . . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅))) |
24 | 21, 23 | mtbiri 330 | . 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁)) |
25 | 24 | necon2ai 3016 | 1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ωcom 7560 2oc2o 8079 ∈𝑔cgoe 32693 ∀𝑔cgol 32695 Fmlacfmla 32697 |
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-inf2 9088 |
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-ral 3111 df-rex 3112 df-reu 3113 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-iun 4883 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-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-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-map 8391 df-goel 32700 df-goal 32702 df-sat 32703 df-fmla 32705 |
This theorem is referenced by: goalr 32757 |
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