Theorem isfmlasuc 35356

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 35354	. . . 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 2830	. 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 4176	. . 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 2744	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑢⊼𝑔𝑣) ↔ 𝐹 = (𝑢⊼𝑔𝑣)))
7	6	rexbidv 3185	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑣 ∈ (Fmla‘𝑁)𝑓 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣)))
8		eqeq1 2744	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 = ∀𝑔𝑖𝑢 ↔ 𝐹 = ∀𝑔𝑖𝑢))
9	8	rexbidv 3185	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑖 ∈ ω 𝑓 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))
10	7, 9	orbi12d 917	. . . . . 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 3185	. . . . 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 3690	. . . 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 914	. 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 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974  suc csuc 6397  ‘cfv 6573  (class class class)co 7448  ωcom 7903  ⊼_𝑔cgna 35302  ∀_𝑔cgol 35303  Fmlacfmla 35305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-map 8886  df-goel 35308  df-goal 35310  df-sat 35311  df-fmla 35313

This theorem is referenced by:  gonarlem  35362  gonar  35363  goalrlem  35364  goalr  35365  fmlasucdisj  35367