Theorem isfmlasuc 35590

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 35588	. . . 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 2823	. 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 4094	. . 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 2741	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑢⊼𝑔𝑣) ↔ 𝐹 = (𝑢⊼𝑔𝑣)))
7	6	rexbidv 3162	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣)))
8		eqeq1 2741	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = ∀𝑔𝑖𝑢 ↔ 𝐹 = ∀𝑔𝑖𝑢))
9	8	rexbidv 3162	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))
10	7, 9	orbi12d 919	. . . . . 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 3162	. . . . 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 3620	. . . 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 916	. 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 848   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ∪ cun 3888  suc csuc 6321  ‘cfv 6494  (class class class)co 7362  ωcom 7812  ⊼_𝑔cgna 35536  ∀_𝑔cgol 35537  Fmlacfmla 35539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-map 8770  df-goel 35542  df-goal 35544  df-sat 35545  df-fmla 35547

This theorem is referenced by:  gonarlem  35596  gonar  35597  goalrlem  35598  goalr  35599  fmlasucdisj  35601