Theorem prv1n 35418

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 ∧ 𝑋 ∈ 𝑉) → ¬ {𝑋}⊧(𝐼∈𝑔𝐽))

Proof of Theorem prv1n

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . 6 ⊢ (ω × {𝑋}) = (ω × {𝑋})
2		omex 9596	. . . . . . . 8 ⊢ ω ∈ V
3		snex 5391	. . . . . . . 8 ⊢ {𝑋} ∈ V
4	2, 3	xpex 7729	. . . . . . 7 ⊢ (ω × {𝑋}) ∈ V
5		eqeq1 2733	. . . . . . 7 ⊢ (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
6	4, 5	spcev 3572	. . . . . 6 ⊢ ((ω × {𝑋}) = (ω × {𝑋}) → ∃𝑎 𝑎 = (ω × {𝑋}))
7	1, 6	mp1i 13	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∃𝑎 𝑎 = (ω × {𝑋}))
8	3, 2	pm3.2i 470	. . . . . . . 8 ⊢ ({𝑋} ∈ V ∧ ω ∈ V)
9		elmapg 8812	. . . . . . . 8 ⊢ (({𝑋} ∈ V ∧ ω ∈ V) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
10	8, 9	mp1i 13	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
11		fconst2g 7177	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
12	11	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
13	10, 12	bitrd 279	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎 = (ω × {𝑋})))
14	13	exbidv 1921	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (∃𝑎 𝑎 ∈ ({𝑋} ↑m ω) ↔ ∃𝑎 𝑎 = (ω × {𝑋})))
15	7, 14	mpbird 257	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
16		neq0 4315	. . . 4 ⊢ (¬ ({𝑋} ↑m ω) = ∅ ↔ ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
17	15, 16	sylibr 234	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ({𝑋} ↑m ω) = ∅)
18		eqcom 2736	. . 3 ⊢ (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
19	17, 18	sylnib 328	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20		ovex 7420	. . . . 5 ⊢ (𝐼∈𝑔𝐽) ∈ V
21	3, 20	pm3.2i 470	. . . 4 ⊢ ({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V)
22		prv 35415	. . . 4 ⊢ (({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω)))
23	21, 22	mp1i 13	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω)))
24		goel 35334	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25		0ex 5262	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
26	25	snid 4626	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28		opelxpi 5675	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
29	27, 28	opelxpd 5677	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
30	24, 29	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ ({∅} × (ω × ω)))
31		fmla0xp 35370	. . . . . . . 8 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
32	30, 31	eleqtrrdi 2839	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
33	32	3adant3 1132	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
34		satefvfmla0 35405	. . . . . 6 ⊢ (({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ (Fmla‘∅)) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽))))})
35	3, 33, 34	sylancr 587	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽))))})
36	24	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37		opex 5424	. . . . . . . . . . . . . 14 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
38	25, 37	op2nd 7977	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽⟩
39	36, 38	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
40	39	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41		op1stg 7980	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
42	40, 41	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐼)
43	42	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐼))
44	39	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45		op2ndg 7981	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
46	44, 45	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐽)
47	46	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐽))
48	43, 47	eleq12d 2822	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
49	48	rabbidv 3413	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)})
50	49	3adant3 1132	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽))))} = {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)})
51		elmapi 8822	. . . . . . . . . 10 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52		elirr 9550	. . . . . . . . . . . 12 ⊢ ¬ 𝑋 ∈ 𝑋
53		fvconst 7136	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎‘𝐼) = 𝑋)
54	53	3ad2antr1 1189	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐼) = 𝑋)
55		fvconst 7136	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎‘𝐽) = 𝑋)
56	55	3ad2antr2 1190	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐽) = 𝑋)
57	54, 56	eleq12d 2822	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → ((𝑎‘𝐼) ∈ (𝑎‘𝐽) ↔ 𝑋 ∈ 𝑋))
58	52, 57	mtbiri 327	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
59	58	ex 412	. . . . . . . . . 10 ⊢ (𝑎:ω⟶{𝑋} → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
60	51, 59	syl 17	. . . . . . . . 9 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
61	60	impcom 407	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) ∧ 𝑎 ∈ ({𝑋} ↑m ω)) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
62	61	ralrimiva 3125	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
63		rabeq0 4351	. . . . . . 7 ⊢ ({𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)} = ∅ ↔ ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
64	62, 63	sylibr 234	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)} = ∅)
65	50, 64	eqtrd 2764	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽))))} = ∅)
66	35, 65	eqtrd 2764	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ∅)
67	66	eqeq1d 2731	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω) ↔ ∅ = ({𝑋} ↑m ω)))
68	23, 67	bitrd 279	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ∅ = ({𝑋} ↑m ω)))
69	19, 68	mtbird 325	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ {𝑋}⊧(𝐼∈𝑔𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  ∅c0 4296  {csn 4589  ⟨cop 4595   class class class wbr 5107   × cxp 5636  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ωcom 7842  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799  ∈_𝑔cgoe 35320  Fmlacfmla 35324   Sat_∈ csate 35325  ⊧cprv 35326

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594  ax-ac2 10416

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-ac 10069  df-goel 35327  df-gona 35328  df-goal 35329  df-sat 35330  df-sate 35331  df-fmla 35332  df-prv 35333

This theorem is referenced by: (None)