Theorem fmla0xp 35577

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 35576	. 2 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
2		rabab 3471	. 2 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
3		eqabcb 2876	. . 3 ⊢ ({𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω)) ↔ ∀𝑥(∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω))))
4		goel 35541	. . . . . 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 3197	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
7		0ex 5252	. . . . . . . . . 10 ⊢ ∅ ∈ V
8	7	snid 4619	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ {∅})
10		opelxpi 5661	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
11	9, 10	opelxpd 5663	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω)))
12		eleq1 2824	. . . . . . 7 ⊢ (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑥 ∈ ({∅} × (ω × ω)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω))))
13	11, 12	syl5ibrcom 247	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω))))
14	13	rexlimivv 3178	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω)))
15		elxpi 5646	. . . . . 6 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))))
16		elsni 4597	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
17	16	opeq1d 4835	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → ⟨𝑦, 𝑧⟩ = ⟨∅, 𝑧⟩)
18	17	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
20		elxpi 5646	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ω × ω) → ∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)))
21		simprr 772	. . . . . . . . . . . . . . 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 4830	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 1920	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨∅, 𝑧⟩ → (∃𝑖∃𝑗(𝑧 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)))
30		r2ex 3173	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ ∃𝑖∃𝑗((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
31	29, 30	imbitrrdi 252	. . . . . . . . . . 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 240	. . . . . . . 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 1933	. . . . . 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 209	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ ↔ 𝑥 ∈ ({∅} × (ω × ω)))
39	6, 38	bitri 275	. . 3 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω)))
40	3, 39	mpgbir 1800	. 2 ⊢ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω))
41	1, 2, 40	3eqtri 2763	1 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∃wrex 3060  {crab 3399  Vcvv 3440  ∅c0 4285  {csn 4580  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358  ωcom 7808  ∈_𝑔cgoe 35527  Fmlacfmla 35531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-map 8765  df-goel 35534  df-sat 35537  df-fmla 35539

This theorem is referenced by:  fmla1  35581  satefvfmla0  35612  prv1n  35625