Theorem fmla0xp 33977

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 33976	. 2 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
2		rabab 3473	. 2 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
3		eqabc 2879	. . 3 ⊢ ({𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω)) ↔ ∀𝑥(∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω))))
4		goel 33941	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
5	4	eqeq2d 2747	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
6	5	2rexbiia 3209	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
7		0ex 5264	. . . . . . . . . 10 ⊢ ∅ ∈ V
8	7	snid 4622	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ {∅})
10		opelxpi 5670	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
11	9, 10	opelxpd 5671	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω)))
12		eleq1 2825	. . . . . . 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 3196	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω)))
15		elxpi 5655	. . . . . 6 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))))
16		elsni 4603	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
17	16	opeq1d 4836	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → ⟨𝑦, 𝑧⟩ = ⟨∅, 𝑧⟩)
18	17	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
20		elxpi 5655	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ω × ω) → ∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)))
21		simprr 771	. . . . . . . . . . . . . . 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 4831	. . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 1922	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → (∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
30		r2ex 3192	. . . . . . . . . . . 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 482	. . . . . . . . 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 408	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
36	35	exlimivv 1935	. . . . . 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 1801	. 2 ⊢ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω))
41	1, 2, 40	3eqtri 2768	1 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∃wrex 3073  {crab 3407  Vcvv 3445  ∅c0 4282  {csn 4586  ⟨cop 4592   × cxp 5631  ‘cfv 6496  (class class class)co 7357  ωcom 7802  ∈_𝑔cgoe 33927  Fmlacfmla 33931

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-map 8767  df-goel 33934  df-sat 33937  df-fmla 33939

This theorem is referenced by:  fmla1  33981  satefvfmla0  34012  prv1n  34025