Theorem prv1n 35425

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 2730	. . . . . 6 ⊢ (ω × {𝑋}) = (ω × {𝑋})
2		omex 9603	. . . . . . . 8 ⊢ ω ∈ V
3		snex 5394	. . . . . . . 8 ⊢ {𝑋} ∈ V
4	2, 3	xpex 7732	. . . . . . 7 ⊢ (ω × {𝑋}) ∈ V
5		eqeq1 2734	. . . . . . 7 ⊢ (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
6	4, 5	spcev 3575	. . . . . 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 8815	. . . . . . . 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 7180	. . . . . . . 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 4318	. . . 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 2737	. . 3 ⊢ (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
19	17, 18	sylnib 328	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20		ovex 7423	. . . . 5 ⊢ (𝐼∈𝑔𝐽) ∈ V
21	3, 20	pm3.2i 470	. . . 4 ⊢ ({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V)
22		prv 35422	. . . 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 35341	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25		0ex 5265	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
26	25	snid 4629	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28		opelxpi 5678	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
29	27, 28	opelxpd 5680	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
30	24, 29	eqeltrd 2829	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ ({∅} × (ω × ω)))
31		fmla0xp 35377	. . . . . . . 8 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
32	30, 31	eleqtrrdi 2840	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
33	32	3adant3 1132	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
34		satefvfmla0 35412	. . . . . 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 6865	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37		opex 5427	. . . . . . . . . . . . . 14 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
38	25, 37	op2nd 7980	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽⟩
39	36, 38	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
40	39	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41		op1stg 7983	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
42	40, 41	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐼)
43	42	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐼))
44	39	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45		op2ndg 7984	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
46	44, 45	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐽)
47	46	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐽))
48	43, 47	eleq12d 2823	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
49	48	rabbidv 3416	. . . . . . 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 8825	. . . . . . . . . 10 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52		elirr 9557	. . . . . . . . . . . 12 ⊢ ¬ 𝑋 ∈ 𝑋
53		fvconst 7139	. . . . . . . . . . . . . 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 7139	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . 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 3126	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
63		rabeq0 4354	. . . . . . 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 2765	. . . . 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 2765	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ∅)
67	66	eqeq1d 2732	. . 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 3045  {crab 3408  Vcvv 3450  ∅c0 4299  {csn 4592  ⟨cop 4598   class class class wbr 5110   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ωcom 7845  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802  ∈_𝑔cgoe 35327  Fmlacfmla 35331   Sat_∈ csate 35332  ⊧cprv 35333

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601  ax-ac2 10423

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-ac 10076  df-goel 35334  df-gona 35335  df-goal 35336  df-sat 35337  df-sate 35338  df-fmla 35339  df-prv 35340

This theorem is referenced by: (None)