Theorem satf0n0 35405

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 6881	. . . . 5 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
2	1	eleq2d 2821	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘∅)))
3	2	notbid 318	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)))
4		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
5	4	eleq2d 2821	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
6	5	notbid 318	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
7		fveq2 6881	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
8	7	eleq2d 2821	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
9	8	notbid 318	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
10		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
11	10	eleq2d 2821	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
12	11	notbid 318	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
13		0nelopab 5547	. . . 4 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
14		satf00 35401	. . . . 5 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
15	14	eleq2i 2827	. . . 4 ⊢ (∅ ∈ ((∅ Sat ∅)‘∅) ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})
16	13, 15	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)
17		simpr 484	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦))
18		0nelopab 5547	. . . . . 6 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}
19		ioran 985	. . . . . 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 590	. . . . 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 2736	. . . . . . . . 9 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
22	21	satf0suc 35403	. . . . . . . 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 480	. . . . . . 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 2821	. . . . . 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 4133	. . . . . 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 287	. . . . 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 325	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦))
28	27	ex 412	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
29	3, 6, 9, 12, 16, 28	finds 7897	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
30		df-nel 3038	. 2 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
31	29, 30	sylibr 234	1 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∉ wnel 3037  ∃wrex 3061   ∪ cun 3929  ∅c0 4313  {copab 5186  suc csuc 6359  ‘cfv 6536  (class class class)co 7410  ωcom 7866  1^st c1st 7991  ∈_𝑔cgoe 35360  ⊼_𝑔cgna 35361  ∀_𝑔cgol 35362   Sat csat 35363

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-map 8847  df-goel 35367  df-sat 35370

This theorem is referenced by:  fmlafvel  35412