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Theorem satf0n0 34369
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.
StepHypRef Expression
1 fveq2 6892 . . . . 5 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
21eleq2d 2820 . . . 4 (𝑥 = ∅ → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘∅)))
32notbid 318 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘∅)))
4 fveq2 6892 . . . . 5 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
54eleq2d 2820 . . . 4 (𝑥 = 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
65notbid 318 . . 3 (𝑥 = 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)))
7 fveq2 6892 . . . . 5 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
87eleq2d 2820 . . . 4 (𝑥 = suc 𝑦 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
98notbid 318 . . 3 (𝑥 = suc 𝑦 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
10 fveq2 6892 . . . . 5 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
1110eleq2d 2820 . . . 4 (𝑥 = 𝑁 → (∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
1211notbid 318 . . 3 (𝑥 = 𝑁 → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑥) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁)))
13 0nelopab 5568 . . . 4 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
14 satf00 34365 . . . . 5 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
1514eleq2i 2826 . . . 4 (∅ ∈ ((∅ Sat ∅)‘∅) ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
1613, 15mtbir 323 . . 3 ¬ ∅ ∈ ((∅ Sat ∅)‘∅)
17 simpr 486 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦))
18 0nelopab 5568 . . . . . 6 ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}
19 ioran 983 . . . . . 6 (¬ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ↔ (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) ∧ ¬ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
2017, 18, 19sylanblrc 591 . . . . 5 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
21 eqid 2733 . . . . . . . . 9 (∅ Sat ∅) = (∅ Sat ∅)
2221satf0suc 34367 . . . . . . . 8 (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
2322adantr 482 . . . . . . 7 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
2423eleq2d 2820 . . . . . 6 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → (∅ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ ∅ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
25 elun 4149 . . . . . 6 (∅ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ↔ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
2624, 25bitrdi 287 . . . . 5 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → (∅ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (∅ ∈ ((∅ Sat ∅)‘𝑦) ∨ ∅ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
2720, 26mtbird 325 . . . 4 ((𝑦 ∈ ω ∧ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑦)) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦))
2827ex 414 . . 3 (𝑦 ∈ ω → (¬ ∅ ∈ ((∅ Sat ∅)‘𝑦) → ¬ ∅ ∈ ((∅ Sat ∅)‘suc 𝑦)))
293, 6, 9, 12, 16, 28finds 7889 . 2 (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
30 df-nel 3048 . 2 (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
3129, 30sylibr 233 1 (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wnel 3047  wrex 3071  cun 3947  c0 4323  {copab 5211  suc csuc 6367  cfv 6544  (class class class)co 7409  ωcom 7855  1st c1st 7973  𝑔cgoe 34324  𝑔cgna 34325  𝑔cgol 34326   Sat csat 34327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-map 8822  df-goel 34331  df-sat 34334
This theorem is referenced by:  fmlafvel  34376
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