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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 32610 | . . . 4 ⊢ ∀𝑔𝑖𝐴 = 〈2o, 〈𝑖, 𝐴〉〉 | |
2 | 2on0 8106 | . . . . . . . . . . . 12 ⊢ 2o ≠ ∅ | |
3 | 2 | neii 3017 | . . . . . . . . . . 11 ⊢ ¬ 2o = ∅ |
4 | 3 | intnanr 490 | . . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉) |
5 | 2oex 8105 | . . . . . . . . . . 11 ⊢ 2o ∈ V | |
6 | opex 5349 | . . . . . . . . . . 11 ⊢ 〈𝑖, 𝐴〉 ∈ V | |
7 | 5, 6 | opth 5361 | . . . . . . . . . 10 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 ↔ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉)) |
8 | 4, 7 | mtbir 325 | . . . . . . . . 9 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 |
9 | goel 32615 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = 〈∅, 〈𝑘, 𝑗〉〉) | |
10 | 9 | eqeq2d 2831 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉)) |
11 | 8, 10 | mtbiri 329 | . . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
12 | 11 | rgen2 3202 | . . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
13 | ralnex2 3259 | . . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) | |
14 | 12, 13 | mpbi 232 | . . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
15 | 14 | intnan 489 | . . . . 5 ⊢ ¬ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
16 | eqeq1 2824 | . . . . . . 7 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) | |
17 | 16 | 2rexbidv 3299 | . . . . . 6 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
18 | fmla0 32650 | . . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)} | |
19 | 17, 18 | elrab2 3679 | . . . . 5 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) ↔ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
20 | 15, 19 | mtbir 325 | . . . 4 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) |
21 | 1, 20 | eqneltri 2905 | . . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅) |
22 | fveq2 6663 | . . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅)) | |
23 | 22 | eleq2d 2897 | . . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅))) |
24 | 21, 23 | mtbiri 329 | . 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁)) |
25 | 24 | necon2ai 3044 | 1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 Vcvv 3491 ∅c0 4284 〈cop 4566 ‘cfv 6348 (class class class)co 7149 ωcom 7573 2oc2o 8089 ∈𝑔cgoe 32601 ∀𝑔cgol 32603 Fmlacfmla 32605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-map 8401 df-goel 32608 df-goal 32610 df-sat 32611 df-fmla 32613 |
This theorem is referenced by: goalr 32665 |
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