Theorem satf0n0 32625

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 6670	. . . . 5 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
2	1	eleq2d 2898	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘∅)))
3	2	notbid 320	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)))
4		fveq2 6670	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
5	4	eleq2d 2898	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
6	5	notbid 320	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
7		fveq2 6670	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
8	7	eleq2d 2898	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
9	8	notbid 320	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
10		fveq2 6670	. . . . 5 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
11	10	eleq2d 2898	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
12	11	notbid 320	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
13		0nelopab 5452	. . . 4 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
14		satf00 32621	. . . . 5 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
15	14	eleq2i 2904	. . . 4 ⊢ (∅ ∈ ((∅ Sat ∅)‘∅) ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})
16	13, 15	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)
17		simpr 487	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦))
18		0nelopab 5452	. . . . . 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 592	. . . . 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 2821	. . . . . . . . 9 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
22	21	satf0suc 32623	. . . . . . . 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 483	. . . . . . 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 2898	. . . . . 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 4125	. . . . . 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	syl6bb 289	. . . . 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 327	. . . 4 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦))
28	27	ex 415	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
29	3, 6, 9, 12, 16, 28	finds 7608	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
30		df-nel 3124	. 2 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
31	29, 30	sylibr 236	1 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ∉ wnel 3123  ∃wrex 3139   ∪ cun 3934  ∅c0 4291  {copab 5128  suc csuc 6193  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1^st c1st 7687  ∈_𝑔cgoe 32580  ⊼_𝑔cgna 32581  ∀_𝑔cgol 32582   Sat csat 32583

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-map 8408  df-goel 32587  df-sat 32590

This theorem is referenced by:  fmlafvel  32632