Theorem fmla0xp 33245

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 33244	. 2 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
2		rabab 3450	. 2 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
3		abeq1 2872	. . 3 ⊢ ({𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω)) ↔ ∀𝑥(∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω))))
4		goel 33209	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
5	4	eqeq2d 2749	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
6	5	2rexbiia 3226	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
7		0ex 5226	. . . . . . . . . 10 ⊢ ∅ ∈ V
8	7	snid 4594	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ {∅})
10		opelxpi 5617	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
11	9, 10	opelxpd 5618	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω)))
12		eleq1 2826	. . . . . . 7 ⊢ (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑥 ∈ ({∅} × (ω × ω)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω))))
13	11, 12	syl5ibrcom 246	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω))))
14	13	rexlimivv 3220	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω)))
15		elxpi 5602	. . . . . 6 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))))
16		elsni 4575	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
17	16	opeq1d 4807	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → ⟨𝑦, 𝑧⟩ = ⟨∅, 𝑧⟩)
18	17	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
20		elxpi 5602	. . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → 𝑥 = ⟨∅, 𝑧⟩)
23		opeq2 4802	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
24	23	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
25	24	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → ⟨∅, 𝑧⟩ = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
26	22, 25	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
27	21, 26	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ⟨∅, 𝑧⟩ ∧ (𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω))) → ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
28	27	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → ((𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
29	28	2eximdv 1923	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → (∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
30		r2ex 3231	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
31	29, 30	syl6ibr 251	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨∅, 𝑧⟩ → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
34	19, 33	sylbid 239	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
35	34	impcom 407	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
36	35	exlimivv 1936	. . . . . 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 208	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ 𝑥 ∈ ({∅} × (ω × ω)))
39	6, 38	bitri 274	. . 3 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω)))
40	3, 39	mpgbir 1803	. 2 ⊢ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω))
41	1, 2, 40	3eqtri 2770	1 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {crab 3067  Vcvv 3422  ∅c0 4253  {csn 4558  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255  ωcom 7687  ∈_𝑔cgoe 33195  Fmlacfmla 33199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-map 8575  df-goel 33202  df-sat 33205  df-fmla 33207

This theorem is referenced by:  fmla1  33249  satefvfmla0  33280  prv1n  33293