Theorem satf0n0 35725

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 6867	. . . . 5 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
2	1	eleq2d 2848	. . . 4 ⊢ (𝑥 = ∅ → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘∅)))
3	2	notbid 320	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)))
4		fveq2 6867	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
5	4	eleq2d 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
6	5	notbid 320	. . 3 ⊢ (𝑥 = 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
7		fveq2 6867	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
8	7	eleq2d 2848	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
9	8	notbid 320	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
10		fveq2 6867	. . . . 5 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
11	10	eleq2d 2848	. . . 4 ⊢ (𝑥 = 𝑁 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
12	11	notbid 320	. . 3 ⊢ (𝑥 = 𝑁 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
13		0nelopab 5536	. . . 4 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
14		satf00 35721	. . . . 5 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
15	14	eleq2i 2854	. . . 4 ⊢ (∅ ∈ ((∅ Sat ∅)‘∅) ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})
16	13, 15	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)
17		simpr 488	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦))
18		0nelopab 5536	. . . . . 6 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}
19		ioran 997	. . . . . 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 599	. . . . 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 2762	. . . . . . . . 9 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
22	21	satf0suc 35723	. . . . . . . 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 484	. . . . . . 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 2848	. . . . . 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 4106	. . . . . 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 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 416	. . 3 ⊢ (𝑦 ∈ ω → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
29	3, 6, 9, 12, 16, 28	finds 7877	. 2 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
30		df-nel 3062	. 2 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
31	29, 30	sylibr 236	1 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ∉ wnel 3061  ∃wrex 3086   ∪ cun 3902  ∅c0 4285  {copab 5162  suc csuc 6348  ‘cfv 6521  (class class class)co 7396  ωcom 7846  1^st c1st 7968  ∈_𝑔cgoe 35680  ⊼_𝑔cgna 35681  ∀_𝑔cgol 35682   Sat csat 35683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-map 8810  df-goel 35687  df-sat 35690

This theorem is referenced by:  fmlafvel  35732