![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
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 34264 | . . . 4 ⊢ ∀𝑔𝑖𝐴 = 〈2o, 〈𝑖, 𝐴〉〉 | |
2 | 2on0 8469 | . . . . . . . . . . . 12 ⊢ 2o ≠ ∅ | |
3 | 2 | neii 2943 | . . . . . . . . . . 11 ⊢ ¬ 2o = ∅ |
4 | 3 | intnanr 489 | . . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉) |
5 | 2oex 8464 | . . . . . . . . . . 11 ⊢ 2o ∈ V | |
6 | opex 5460 | . . . . . . . . . . 11 ⊢ 〈𝑖, 𝐴〉 ∈ V | |
7 | 5, 6 | opth 5472 | . . . . . . . . . 10 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 ↔ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉)) |
8 | 4, 7 | mtbir 323 | . . . . . . . . 9 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 |
9 | goel 34269 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = 〈∅, 〈𝑘, 𝑗〉〉) | |
10 | 9 | eqeq2d 2744 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉)) |
11 | 8, 10 | mtbiri 327 | . . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
12 | 11 | rgen2 3198 | . . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
13 | ralnex2 3134 | . . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) | |
14 | 12, 13 | mpbi 229 | . . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
15 | 14 | intnan 488 | . . . . 5 ⊢ ¬ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
16 | eqeq1 2737 | . . . . . . 7 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) | |
17 | 16 | 2rexbidv 3220 | . . . . . 6 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
18 | fmla0 34304 | . . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)} | |
19 | 17, 18 | elrab2 3684 | . . . . 5 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) ↔ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) |
20 | 15, 19 | mtbir 323 | . . . 4 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 ∈ (Fmla‘∅) |
21 | 1, 20 | eqneltri 2853 | . . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅) |
22 | fveq2 6881 | . . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅)) | |
23 | 22 | eleq2d 2820 | . . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅))) |
24 | 21, 23 | mtbiri 327 | . 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁)) |
25 | 24 | necon2ai 2971 | 1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∅c0 4320 〈cop 4630 ‘cfv 6535 (class class class)co 7396 ωcom 7842 2oc2o 8447 ∈𝑔cgoe 34255 ∀𝑔cgol 34257 Fmlacfmla 34259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-map 8810 df-goel 34262 df-goal 34264 df-sat 34265 df-fmla 34267 |
This theorem is referenced by: goalr 34319 |
Copyright terms: Public domain | W3C validator |