Theorem fmla0xp 32245

Description: The valid Godel formulas of height 0 is the set of all formulas of the form v_i ∈ v_j ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)

Assertion

Ref	Expression
fmla0xp	⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Proof of Theorem fmla0xp

Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmla0 32244	. 2 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
2		rabab 3466	. 2 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
3		abeq1 2915	. . 3 ⊢ ({𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω)) ↔ ∀𝑥(∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω))))
4		goel 32209	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
5	4	eqeq2d 2805	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
6	5	2rexbiia 3261	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
7		0ex 5107	. . . . . . . . . 10 ⊢ ∅ ∈ V
8	7	snid 4510	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ {∅})
10		opelxpi 5485	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
11	9, 10	opelxpd 5486	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω)))
12		eleq1 2870	. . . . . . 7 ⊢ (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑥 ∈ ({∅} × (ω × ω)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω))))
13	11, 12	syl5ibrcom 248	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω))))
14	13	rexlimivv 3255	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω)))
15		elxpi 5470	. . . . . 6 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))))
16		elsni 4493	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
17	16	opeq1d 4720	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → ⟨𝑦, 𝑧⟩ = ⟨∅, 𝑧⟩)
18	17	eqeq2d 2805	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
20		elxpi 5470	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ω × ω) → ∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)))
21		simprr 769	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → (𝑖 ∈ ω ∧ 𝑗 ∈ ω))
22		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → 𝑥 = ⟨∅, 𝑧⟩)
23		opeq2 4715	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
24	23	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
25	24	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
26	22, 25	eqtrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
27	21, 26	jca 512	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
28	27	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → ((𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
29	28	2eximdv 1897	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → (∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
30		r2ex 3266	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
31	29, 30	syl6ibr 253	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → (∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
32	20, 31	syl5com 31	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ω × ω) → (𝑥 = ⟨∅, 𝑧⟩ → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
33	32	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨∅, 𝑧⟩ → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
34	19, 33	sylbid 241	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
35	34	impcom 408	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
36	35	exlimivv 1910	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
37	15, 36	syl 17	. . . . 5 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
38	14, 37	impbii 210	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ 𝑥 ∈ ({∅} × (ω × ω)))
39	6, 38	bitri 276	. . 3 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω)))
40	3, 39	mpgbir 1781	. 2 ⊢ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω))
41	1, 2, 40	3eqtri 2823	1 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∃wrex 3106  {crab 3109  Vcvv 3437  ∅c0 4215  {csn 4476  ⟨cop 4482   × cxp 5446  ‘cfv 6230  (class class class)co 7021  ωcom 7441  ∈_𝑔cgoe 32195  Fmlacfmla 32199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-inf2 8955

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-map 8263  df-goel 32202  df-sat 32205  df-fmla 32207

This theorem is referenced by:  fmla1  32249  satefvfmla0  32280  prv1n  32293