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Theorem prv1n 35436
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 2737 . . . . . 6 (ω × {𝑋}) = (ω × {𝑋})
2 omex 9683 . . . . . . . 8 ω ∈ V
3 snex 5436 . . . . . . . 8 {𝑋} ∈ V
42, 3xpex 7773 . . . . . . 7 (ω × {𝑋}) ∈ V
5 eqeq1 2741 . . . . . . 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 8879 . . . . . . . 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 1136 . . . . . . 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 1921 . . . . 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 4352 . . . 4 (¬ ({𝑋} ↑m ω) = ∅ ↔ ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
1715, 16sylibr 234 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ ({𝑋} ↑m ω) = ∅)
18 eqcom 2744 . . 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 35433 . . . 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 35352 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
2625snid 4662 . . . . . . . . . . 11 ∅ ∈ {∅}
2726a1i 11 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28 opelxpi 5722 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
2927, 28opelxpd 5724 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
3024, 29eqeltrd 2841 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ ({∅} × (ω × ω)))
31 fmla0xp 35388 . . . . . . . 8 (Fmla‘∅) = ({∅} × (ω × ω))
3230, 31eleqtrrdi 2852 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
33323adant3 1133 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
34 satefvfmla0 35423 . . . . . 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 6910 . . . . . . . . . . . . 13 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37 opex 5469 . . . . . . . . . . . . . 14 𝐼, 𝐽⟩ ∈ V
3825, 37op2nd 8023 . . . . . . . . . . . . 13 (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽
3936, 38eqtrdi 2793 . . . . . . . . . . . 12 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
4039fveq2d 6910 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41 op1stg 8026 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
4240, 41eqtrd 2777 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐼)
4342fveq2d 6910 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐼))
4439fveq2d 6910 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45 op2ndg 8027 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
4644, 45eqtrd 2777 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐽)
4746fveq2d 6910 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐽))
4843, 47eleq12d 2835 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
4948rabbidv 3444 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)})
50493adant3 1133 . . . . . 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 8889 . . . . . . . . . 10 (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52 elirr 9637 . . . . . . . . . . . 12 ¬ 𝑋𝑋
53 fvconst 7184 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎𝐼) = 𝑋)
54533ad2antr1 1189 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐼) = 𝑋)
55 fvconst 7184 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎𝐽) = 𝑋)
56553ad2antr2 1190 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐽) = 𝑋)
5754, 56eleq12d 2835 . . . . . . . . . . . 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 3146 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎𝐼) ∈ (𝑎𝐽))
63 rabeq0 4388 . . . . . . 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 2777 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = ∅)
6635, 65eqtrd 2777 . . . 4 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋} Sat (𝐼𝑔𝐽)) = ∅)
6766eqeq1d 2739 . . 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 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  c0 4333  {csn 4626  cop 4632   class class class wbr 5143   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  2nd c2nd 8013  m cmap 8866  𝑔cgoe 35338  Fmlacfmla 35342   Sat csate 35343  cprv 35344
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-ac 10156  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-sate 35349  df-fmla 35350  df-prv 35351
This theorem is referenced by: (None)
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