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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 35327 | . . . 4 ⊢ ∀𝑔𝑖𝐴 = 〈2o, 〈𝑖, 𝐴〉〉 | |
2 | 2on0 8521 | . . . . . . . . . . . 12 ⊢ 2o ≠ ∅ | |
3 | 2 | neii 2940 | . . . . . . . . . . 11 ⊢ ¬ 2o = ∅ |
4 | 3 | intnanr 487 | . . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉) |
5 | 2oex 8516 | . . . . . . . . . . 11 ⊢ 2o ∈ V | |
6 | opex 5475 | . . . . . . . . . . 11 ⊢ 〈𝑖, 𝐴〉 ∈ V | |
7 | 5, 6 | opth 5487 | . . . . . . . . . 10 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 ↔ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉)) |
8 | 4, 7 | mtbir 323 | . . . . . . . . 9 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 |
9 | goel 35332 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = 〈∅, 〈𝑘, 𝑗〉〉) | |
10 | 9 | eqeq2d 2746 | . . . . . . . . 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 3197 | . . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
13 | ralnex2 3131 | . . . . . . 7 ⊢ (∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) ↔ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) | |
14 | 12, 13 | mpbi 230 | . . . . . 6 ⊢ ¬ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) |
15 | 14 | intnan 486 | . . . . 5 ⊢ ¬ (〈2o, 〈𝑖, 𝐴〉〉 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗)) |
16 | eqeq1 2739 | . . . . . . 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 35367 | . . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)} | |
19 | 17, 18 | elrab2 3698 | . . . . 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 2858 | . . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅) |
22 | fveq2 6907 | . . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅)) | |
23 | 22 | eleq2d 2825 | . . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅))) |
24 | 21, 23 | mtbiri 327 | . 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁)) |
25 | 24 | necon2ai 2968 | 1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∅c0 4339 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ωcom 7887 2oc2o 8499 ∈𝑔cgoe 35318 ∀𝑔cgol 35320 Fmlacfmla 35322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-map 8867 df-goel 35325 df-goal 35327 df-sat 35328 df-fmla 35330 |
This theorem is referenced by: goalr 35382 |
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