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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 35347 | . . . 4 ⊢ ∀𝑔𝑖𝐴 = 〈2o, 〈𝑖, 𝐴〉〉 | |
| 2 | 2on0 8522 | . . . . . . . . . . . 12 ⊢ 2o ≠ ∅ | |
| 3 | 2 | neii 2942 | . . . . . . . . . . 11 ⊢ ¬ 2o = ∅ | 
| 4 | 3 | intnanr 487 | . . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉) | 
| 5 | 2oex 8517 | . . . . . . . . . . 11 ⊢ 2o ∈ V | |
| 6 | opex 5469 | . . . . . . . . . . 11 ⊢ 〈𝑖, 𝐴〉 ∈ V | |
| 7 | 5, 6 | opth 5481 | . . . . . . . . . 10 ⊢ (〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 ↔ (2o = ∅ ∧ 〈𝑖, 𝐴〉 = 〈𝑘, 𝑗〉)) | 
| 8 | 4, 7 | mtbir 323 | . . . . . . . . 9 ⊢ ¬ 〈2o, 〈𝑖, 𝐴〉〉 = 〈∅, 〈𝑘, 𝑗〉〉 | 
| 9 | goel 35352 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑗 ∈ ω) → (𝑘∈𝑔𝑗) = 〈∅, 〈𝑘, 𝑗〉〉) | |
| 10 | 9 | eqeq2d 2748 | . . . . . . . . 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 3199 | . . . . . . 7 ⊢ ∀𝑘 ∈ ω ∀𝑗 ∈ ω ¬ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗) | 
| 13 | ralnex2 3133 | . . . . . . 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 2741 | . . . . . . 7 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) | |
| 17 | 16 | 2rexbidv 3222 | . . . . . 6 ⊢ (𝑥 = 〈2o, 〈𝑖, 𝐴〉〉 → (∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗) ↔ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 〈2o, 〈𝑖, 𝐴〉〉 = (𝑘∈𝑔𝑗))) | 
| 18 | fmla0 35387 | . . . . . 6 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑘∈𝑔𝑗)} | |
| 19 | 17, 18 | elrab2 3695 | . . . . 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 2860 | . . 3 ⊢ ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅) | 
| 22 | fveq2 6906 | . . . 4 ⊢ (𝑁 = ∅ → (Fmla‘𝑁) = (Fmla‘∅)) | |
| 23 | 22 | eleq2d 2827 | . . 3 ⊢ (𝑁 = ∅ → (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) ↔ ∀𝑔𝑖𝐴 ∈ (Fmla‘∅))) | 
| 24 | 21, 23 | mtbiri 327 | . 2 ⊢ (𝑁 = ∅ → ¬ ∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁)) | 
| 25 | 24 | necon2ai 2970 | 1 ⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 ωcom 7887 2oc2o 8500 ∈𝑔cgoe 35338 ∀𝑔cgol 35340 Fmlacfmla 35342 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-map 8868 df-goel 35345 df-goal 35347 df-sat 35348 df-fmla 35350 | 
| This theorem is referenced by: goalr 35402 | 
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