Theorem isfmlasuc 35531

Description: The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)

Assertion

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))))

Distinct variable groups:   𝑖,𝐹,𝑢,𝑣   𝑖,𝑁,𝑢,𝑣

Allowed substitution hints:   𝑉(𝑣,𝑢,𝑖)

Proof of Theorem isfmlasuc

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmlasuc 35529	. . . 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 480	. . 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 2820	. 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 4103	. . 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 2738	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑢⊼𝑔𝑣) ↔ 𝐹 = (𝑢⊼𝑔𝑣)))
7	6	rexbidv 3158	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣)))
8		eqeq1 2738	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = ∀𝑔𝑖𝑢 ↔ 𝐹 = ∀𝑔𝑖𝑢))
9	8	rexbidv 3158	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))
10	7, 9	orbi12d 918	. . . . . 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 3158	. . . . 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 3629	. . . 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 481	. . 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 915	. 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 305	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))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {cab 2712  ∃wrex 3058   ∪ cun 3897  suc csuc 6317  ‘cfv 6490  (class class class)co 7356  ωcom 7806  ⊼_𝑔cgna 35477  ∀_𝑔cgol 35478  Fmlacfmla 35480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-map 8763  df-goel 35483  df-goal 35485  df-sat 35486  df-fmla 35488

This theorem is referenced by:  gonarlem  35537  gonar  35538  goalrlem  35539  goalr  35540  fmlasucdisj  35542