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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfmlasuc | Structured version Visualization version GIF version |
Description: The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.) |
Ref | Expression |
---|---|
isfmlasuc | ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmlasuc 32652 | . . . 4 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)})) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)})) |
3 | 2 | eleq2d 2897 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ 𝐹 ∈ ((Fmla‘𝑁) ∪ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)}))) |
4 | elun 4118 | . . 3 ⊢ (𝐹 ∈ ((Fmla‘𝑁) ∪ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)}) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ 𝐹 ∈ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)})) | |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ ((Fmla‘𝑁) ∪ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)}) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ 𝐹 ∈ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)}))) |
6 | eqeq1 2824 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑢⊼𝑔𝑣) ↔ 𝐹 = (𝑢⊼𝑔𝑣))) | |
7 | 6 | rexbidv 3296 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣))) |
8 | eqeq1 2824 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = ∀𝑔𝑖𝑢 ↔ 𝐹 = ∀𝑔𝑖𝑢)) | |
9 | 8 | rexbidv 3296 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢)) |
10 | 7, 9 | orbi12d 915 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))) |
11 | 10 | rexbidv 3296 | . . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢) ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))) |
12 | 11 | elabg 3662 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)} ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)} ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))) |
14 | 13 | orbi2d 912 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → ((𝐹 ∈ (Fmla‘𝑁) ∨ 𝐹 ∈ {𝑓 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢)}) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢)))) |
15 | 3, 5, 14 | 3bitrd 307 | 1 ⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 {cab 2798 ∃wrex 3138 ∪ cun 3927 suc csuc 6186 ‘cfv 6348 (class class class)co 7149 ωcom 7573 ⊼𝑔cgna 32600 ∀𝑔cgol 32601 Fmlacfmla 32603 |
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-nel 3123 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-map 8401 df-goel 32606 df-goal 32608 df-sat 32609 df-fmla 32611 |
This theorem is referenced by: gonarlem 32660 gonar 32661 goalrlem 32662 goalr 32663 fmlasucdisj 32665 |
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