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Theorem prv1n 33134
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 2739 . . . . . 6 (ω × {𝑋}) = (ω × {𝑋})
2 omex 9285 . . . . . . . 8 ω ∈ V
3 snex 5340 . . . . . . . 8 {𝑋} ∈ V
42, 3xpex 7559 . . . . . . 7 (ω × {𝑋}) ∈ V
5 eqeq1 2743 . . . . . . 7 (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
64, 5spcev 3535 . . . . . 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 474 . . . . . . . 8 ({𝑋} ∈ V ∧ ω ∈ V)
9 elmapg 8544 . . . . . . . 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 7039 . . . . . . . 8 (𝑋𝑉 → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
12113ad2ant3 1137 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
1310, 12bitrd 282 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎 = (ω × {𝑋})))
1413exbidv 1929 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (∃𝑎 𝑎 ∈ ({𝑋} ↑m ω) ↔ ∃𝑎 𝑎 = (ω × {𝑋})))
157, 14mpbird 260 . . . 4 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
16 neq0 4276 . . . 4 (¬ ({𝑋} ↑m ω) = ∅ ↔ ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
1715, 16sylibr 237 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ ({𝑋} ↑m ω) = ∅)
18 eqcom 2746 . . 3 (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
1917, 18sylnib 331 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20 ovex 7267 . . . . 5 (𝐼𝑔𝐽) ∈ V
213, 20pm3.2i 474 . . . 4 ({𝑋} ∈ V ∧ (𝐼𝑔𝐽) ∈ V)
22 prv 33131 . . . 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 33050 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25 0ex 5216 . . . . . . . . . . . 12 ∅ ∈ V
2625snid 4593 . . . . . . . . . . 11 ∅ ∈ {∅}
2726a1i 11 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28 opelxpi 5605 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
2927, 28opelxpd 5606 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
3024, 29eqeltrd 2840 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ ({∅} × (ω × ω)))
31 fmla0xp 33086 . . . . . . . 8 (Fmla‘∅) = ({∅} × (ω × ω))
3230, 31eleqtrrdi 2851 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
33323adant3 1134 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (𝐼𝑔𝐽) ∈ (Fmla‘∅))
34 satefvfmla0 33121 . . . . . 6 (({𝑋} ∈ V ∧ (𝐼𝑔𝐽) ∈ (Fmla‘∅)) → ({𝑋} Sat (𝐼𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))})
353, 33, 34sylancr 590 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋} Sat (𝐼𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))})
3624fveq2d 6742 . . . . . . . . . . . . 13 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37 opex 5364 . . . . . . . . . . . . . 14 𝐼, 𝐽⟩ ∈ V
3825, 37op2nd 7791 . . . . . . . . . . . . 13 (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽
3936, 38eqtrdi 2796 . . . . . . . . . . . 12 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
4039fveq2d 6742 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41 op1stg 7794 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
4240, 41eqtrd 2779 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐼)
4342fveq2d 6742 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐼))
4439fveq2d 6742 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45 op2ndg 7795 . . . . . . . . . . 11 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
4644, 45eqtrd 2779 . . . . . . . . . 10 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼𝑔𝐽))) = 𝐽)
4746fveq2d 6742 . . . . . . . . 9 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) = (𝑎𝐽))
4843, 47eleq12d 2834 . . . . . . . 8 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽)))) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
4948rabbidv 3404 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)})
50493adant3 1134 . . . . . 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 8553 . . . . . . . . . 10 (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52 elirr 9240 . . . . . . . . . . . 12 ¬ 𝑋𝑋
53 fvconst 7000 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎𝐼) = 𝑋)
54533ad2antr1 1190 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐼) = 𝑋)
55 fvconst 7000 . . . . . . . . . . . . . 14 ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎𝐽) = 𝑋)
56553ad2antr2 1191 . . . . . . . . . . . . 13 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → (𝑎𝐽) = 𝑋)
5754, 56eleq12d 2834 . . . . . . . . . . . 12 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → ((𝑎𝐼) ∈ (𝑎𝐽) ↔ 𝑋𝑋))
5852, 57mtbiri 330 . . . . . . . . . . 11 ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉)) → ¬ (𝑎𝐼) ∈ (𝑎𝐽))
5958ex 416 . . . . . . . . . 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 411 . . . . . . . 8 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) ∧ 𝑎 ∈ ({𝑋} ↑m ω)) → ¬ (𝑎𝐼) ∈ (𝑎𝐽))
6261ralrimiva 3107 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎𝐼) ∈ (𝑎𝐽))
63 rabeq0 4315 . . . . . . 7 ({𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)} = ∅ ↔ ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎𝐼) ∈ (𝑎𝐽))
6462, 63sylibr 237 . . . . . 6 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎𝐼) ∈ (𝑎𝐽)} = ∅)
6550, 64eqtrd 2779 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼𝑔𝐽))))} = ∅)
6635, 65eqtrd 2779 . . . 4 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋} Sat (𝐼𝑔𝐽)) = ∅)
6766eqeq1d 2741 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → (({𝑋} Sat (𝐼𝑔𝐽)) = ({𝑋} ↑m ω) ↔ ∅ = ({𝑋} ↑m ω)))
6823, 67bitrd 282 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ({𝑋}⊧(𝐼𝑔𝐽) ↔ ∅ = ({𝑋} ↑m ω)))
6919, 68mtbird 328 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ {𝑋}⊧(𝐼𝑔𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2112  wral 3063  {crab 3067  Vcvv 3422  c0 4253  {csn 4557  cop 4563   class class class wbr 5069   × cxp 5566  wf 6396  cfv 6400  (class class class)co 7234  ωcom 7665  1st c1st 7780  2nd c2nd 7781  m cmap 8531  𝑔cgoe 33036  Fmlacfmla 33040   Sat csate 33041  cprv 33042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-reg 9235  ax-inf2 9283  ax-ac2 10104
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-int 4876  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-se 5527  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-isom 6409  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-1st 7782  df-2nd 7783  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-1o 8225  df-2o 8226  df-er 8414  df-map 8533  df-en 8650  df-dom 8651  df-sdom 8652  df-card 9582  df-ac 9757  df-goel 33043  df-gona 33044  df-goal 33045  df-sat 33046  df-sate 33047  df-fmla 33048  df-prv 33049
This theorem is referenced by: (None)
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