Theorem isfmlasuc 35375

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 35373	. . . 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 2814	. 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 4116	. . 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 2733	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑢⊼𝑔𝑣) ↔ 𝐹 = (𝑢⊼𝑔𝑣)))
7	6	rexbidv 3157	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣)))
8		eqeq1 2733	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = ∀𝑔𝑖𝑢 ↔ 𝐹 = ∀𝑔𝑖𝑢))
9	8	rexbidv 3157	. . . . . . 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 3157	. . . . 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 3643	. . . 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 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ∪ cun 3912  suc csuc 6334  ‘cfv 6511  (class class class)co 7387  ωcom 7842  ⊼_𝑔cgna 35321  ∀_𝑔cgol 35322  Fmlacfmla 35324

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-map 8801  df-goel 35327  df-goal 35329  df-sat 35330  df-fmla 35332

This theorem is referenced by:  gonarlem  35381  gonar  35382  goalrlem  35383  goalr  35384  fmlasucdisj  35386