Theorem satf0n0 34667

Description: The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation does not contain the empty set. (Contributed by AV, 19-Sep-2023.)

Assertion

Ref	Expression
satf0n0	⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))

Proof of Theorem satf0n0

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

Step	Hyp	Ref	Expression
1		fveq2 6890	. . . . 5 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
2	1	eleq2d 2817	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘∅)))
3	2	notbid 317	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)))
4		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
5	4	eleq2d 2817	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
6	5	notbid 317	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
7		fveq2 6890	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
8	7	eleq2d 2817	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
9	8	notbid 317	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
10		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
11	10	eleq2d 2817	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
12	11	notbid 317	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
13		0nelopab 5566	. . . 4 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
14		satf00 34663	. . . . 5 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
15	14	eleq2i 2823	. . . 4 ⊢ (∅ ∈ ((∅ Sat ∅)‘∅) ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})
16	13, 15	mtbir 322	. . 3 ⊢ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)
17		simpr 483	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦))
18		0nelopab 5566	. . . . . 6 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}
19		ioran 980	. . . . . 6 ⊢ (¬ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}) ↔ (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) ∧ ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
20	17, 18, 19	sylanblrc 588	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
21		eqid 2730	. . . . . . . . 9 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
22	21	satf0suc 34665	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
23	22	adantr 479	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
24	23	eleq2d 2817	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → (∅ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ ∅ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})))
25		elun 4147	. . . . . 6 ⊢ (∅ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}) ↔ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
26	24, 25	bitrdi 286	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → (∅ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})))
27	20, 26	mtbird 324	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦))
28	27	ex 411	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
29	3, 6, 9, 12, 16, 28	finds 7891	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
30		df-nel 3045	. 2 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
31	29, 30	sylibr 233	1 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ∉ wnel 3044  ∃wrex 3068   ∪ cun 3945  ∅c0 4321  {copab 5209  suc csuc 6365  ‘cfv 6542  (class class class)co 7411  ωcom 7857  1^st c1st 7975  ∈_𝑔cgoe 34622  ⊼_𝑔cgna 34623  ∀_𝑔cgol 34624   Sat csat 34625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-map 8824  df-goel 34629  df-sat 34632

This theorem is referenced by:  fmlafvel  34674