Theorem fmla0xp 35448

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 35447	. 2 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
2		rabab 3468	. 2 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
3		eqabcb 2873	. . 3 ⊢ ({𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = ({∅} × (ω × ω)) ↔ ∀𝑥(∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 ∈ ({∅} × (ω × ω))))
4		goel 35412	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
5	4	eqeq2d 2744	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
6	5	2rexbiia 3194	. . . 4 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
7		0ex 5247	. . . . . . . . . 10 ⊢ ∅ ∈ V
8	7	snid 4614	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ {∅})
10		opelxpi 5656	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
11	9, 10	opelxpd 5658	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ ({∅} × (ω × ω)))
12		eleq1 2821	. . . . . . 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 3175	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → 𝑥 ∈ ({∅} × (ω × ω)))
15		elxpi 5641	. . . . . 6 ⊢ (𝑥 ∈ ({∅} × (ω × ω)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω))))
16		elsni 4592	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
17	16	opeq1d 4830	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → ⟨𝑦, 𝑧⟩ = ⟨∅, 𝑧⟩)
18	17	eqeq2d 2744	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅} ∧ 𝑧 ∈ (ω × ω)) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ 𝑥 = ⟨∅, 𝑧⟩))
20		elxpi 5641	. . . . . . . . . . 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 4825	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . 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 3170	. . . . . . . . . . . 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 2760	1 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∃wrex 3057  {crab 3396  Vcvv 3437  ∅c0 4282  {csn 4575  ⟨cop 4581   × cxp 5617  ‘cfv 6486  (class class class)co 7352  ωcom 7802  ∈_𝑔cgoe 35398  Fmlacfmla 35402

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-map 8758  df-goel 35405  df-sat 35408  df-fmla 35410

This theorem is referenced by:  fmla1  35452  satefvfmla0  35483  prv1n  35496