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Theorem prv1n 35416
Description: No wff encoded as a Godel-set of membership is true in a model with only one element. (Contributed by AV, 19-Nov-2023.)
Assertion
Ref Expression
prv1n ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ {𝑋}⊧(𝐼𝑔𝐽))

Proof of Theorem prv1n
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . . . 6 (ω × {𝑋}) = (ω × {𝑋})
2 omex 9681 . . . . . . . 8 ω ∈ V
3 snex 5442 . . . . . . . 8 {𝑋} ∈ V
42, 3xpex 7772 . . . . . . 7 (ω × {𝑋}) ∈ V
5 eqeq1 2739 . . . . . . 7 (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
64, 5spcev 3606 . . . . . 6 ((ω × {𝑋}) = (ω × {𝑋}) → ∃𝑎 𝑎 = (ω × {𝑋}))
71, 6mp1i 13 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∃𝑎 𝑎 = (ω × {𝑋}))
83, 2pm3.2i 470 . . . . . . . 8 ({𝑋} ∈ V ∧ ω ∈ V)
9 elmapg 8878 . . . . . . . 8 (({𝑋} ∈ V ∧ ω ∈ V) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
108, 9mp1i 13 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
11 fconst2g 7223 . . . . . . . 8 (𝑋𝑉 → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
12113ad2ant3 1134 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
1310, 12bitrd 279 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎 = (ω × {𝑋})))
1413exbidv 1919 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (∃𝑎 𝑎 ∈ ({𝑋} ↑m ω) ↔ ∃𝑎 𝑎 = (ω × {𝑋})))
157, 14mpbird 257 . . . 4 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
16 neq0 4358 . . . 4 (¬ ({𝑋} ↑m ω) = ∅ ↔ ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
1715, 16sylibr 234 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ ({𝑋} ↑m ω) = ∅)
18 eqcom 2742 . . 3 (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
1917, 18sylnib 328 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20 ovex 7464 . . . . 5 (𝐼𝑔𝐽) ∈ V
213, 20pm3.2i 470 . . . 4 ({𝑋} ∈ V ∧ (𝐼𝑔𝐽) ∈ V)
22 prv 35413 . . . 4 (({𝑋} ∈ V ∧ (𝐼𝑔𝐽) ∈ V) → ({𝑋}⊧(𝐼𝑔𝐽) ↔ ({𝑋} Sat (𝐼𝑔𝐽)) = ({𝑋} ↑m ω)))
2321, 22mp1i 13 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋}⊧(𝐼𝑔𝐽) ↔ ({𝑋} Sat (𝐼𝑔𝐽)) = ({𝑋} ↑m ω)))
24 goel 35332 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25 0ex 5313 . . . . . . . . . . . 12 ∅ ∈ V
2625snid 4667 . . . . . . . . . . 11 ∅ ∈ {∅}
2726a1i 11 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28 opelxpi 5726 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
2927, 28opelxpd 5728 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
3024, 29eqeltrd 2839 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ ({∅} × (ω × ω)))
31 fmla0xp 35368 . . . . . . . 8 (Fmla‘∅) = ({∅} × (ω × ω))
3230, 31eleqtrrdi 2850 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
33323adant3 1131 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
34 satefvfmla0 35403 . . . . . 6 (({𝑋} ∈ V ∧ (𝐼𝑔𝐽) ∈ (Fmla‘∅)) → ({𝑋} Sat (𝐼𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))})
353, 33, 34sylancr 587 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋} Sat (𝐼𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))})
3624fveq2d 6911 . . . . . . . . . . . . 13 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37 opex 5475 . . . . . . . . . . . . . 14 𝐼, 𝐽⟩ ∈ V
3825, 37op2nd 8022 . . . . . . . . . . . . 13 (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽
3936, 38eqtrdi 2791 . . . . . . . . . . . 12 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
4039fveq2d 6911 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41 op1stg 8025 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
4240, 41eqtrd 2775 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐼)
4342fveq2d 6911 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐼))
4439fveq2d 6911 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45 op2ndg 8026 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
4644, 45eqtrd 2775 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐽)
4746fveq2d 6911 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐽))
4843, 47eleq12d 2833 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
4948rabbidv 3441 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)})
50493adant3 1131 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)})
51 elmapi 8888 . . . . . . . . . 10 (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52 elirr 9635 . . . . . . . . . . . 12 ¬ 𝑋𝑋
53 fvconst 7184 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎𝐼) = 𝑋)
54533ad2antr1 1187 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐼) = 𝑋)
55 fvconst 7184 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎𝐽) = 𝑋)
56553ad2antr2 1188 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐽) = 𝑋)
5754, 56eleq12d 2833 . . . . . . . . . . . 12 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → ((𝑎𝐼) ∈ (𝑎𝐽) ↔ 𝑋𝑋))
5852, 57mtbiri 327 . . . . . . . . . . 11 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → ¬ (𝑎𝐼) ∈ (𝑎𝐽))
5958ex 412 . . . . . . . . . 10 (𝑎:ω⟶{𝑋} → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ (𝑎𝐼) ∈ (𝑎𝐽)))
6051, 59syl 17 . . . . . . . . 9 (𝑎 ∈ ({𝑋} ↑m ω) → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ (𝑎𝐼) ∈ (𝑎𝐽)))
6160impcom 407 . . . . . . . 8 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) ∧ 𝑎 ∈ ({𝑋} ↑m ω)) → ¬ (𝑎𝐼) ∈ (𝑎𝐽))
6261ralrimiva 3144 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎𝐼) ∈ (𝑎𝐽))
63 rabeq0 4394 . . . . . . 7 ({𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)} = ∅ ↔ ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎𝐼) ∈ (𝑎𝐽))
6462, 63sylibr 234 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)} = ∅)
6550, 64eqtrd 2775 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = ∅)
6635, 65eqtrd 2775 . . . 4 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋} Sat (𝐼𝑔𝐽)) = ∅)
6766eqeq1d 2737 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (({𝑋} Sat (𝐼𝑔𝐽)) = ({𝑋} ↑m ω) ↔ ∅ = ({𝑋} ↑m ω)))
6823, 67bitrd 279 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋}⊧(𝐼𝑔𝐽) ↔ ∅ = ({𝑋} ↑m ω)))
6919, 68mtbird 325 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ {𝑋}⊧(𝐼𝑔𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  c0 4339  {csn 4631  cop 4637   class class class wbr 5148   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  2nd c2nd 8012  m cmap 8865  𝑔cgoe 35318  Fmlacfmla 35322   Sat csate 35323  cprv 35324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-sate 35329  df-fmla 35330  df-prv 35331
This theorem is referenced by: (None)
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