Theorem prv1n 36397

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 2763	. . . . . 6 ⊢ (ω × {𝑋}) = (ω × {𝑋})
2		omex 9679	. . . . . . . 8 ⊢ ω ∈ V
3		snex 5402	. . . . . . . 8 ⊢ {𝑋} ∈ V
4	2, 3	xpex 7797	. . . . . . 7 ⊢ (ω × {𝑋}) ∈ V
5		eqeq1 2767	. . . . . . 7 ⊢ (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
6	4, 5	spcev 3564	. . . . . 6 ⊢ ((ω × {𝑋}) = (ω × {𝑋}) → ∃𝑎 𝑎 = (ω × {𝑋}))
7	1, 6	mp1i 14	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∃𝑎 𝑎 = (ω × {𝑋}))
8	3, 2	pm3.2i 476	. . . . . . . 8 ⊢ ({𝑋} ∈ V ∧ ω ∈ V)
9		elmapg 8895	. . . . . . . 8 ⊢ (({𝑋} ∈ V ∧ ω ∈ V) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
10	8, 9	mp1i 14	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎:ω⟶{𝑋}))
11		fconst2g 7238	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
12	11	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
13	10, 12	bitrd 282	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝑎 ∈ ({𝑋} ↑m ω) ↔ 𝑎 = (ω × {𝑋})))
14	13	exbidv 1956	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (∃𝑎 𝑎 ∈ ({𝑋} ↑m ω) ↔ ∃𝑎 𝑎 = (ω × {𝑋})))
15	7, 14	mpbird 260	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
16		neq0 4302	. . . 4 ⊢ (¬ ({𝑋} ↑m ω) = ∅ ↔ ∃𝑎 𝑎 ∈ ({𝑋} ↑m ω))
17	15, 16	sylibr 237	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ({𝑋} ↑m ω) = ∅)
18		eqcom 2770	. . 3 ⊢ (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
19	17, 18	sylnib 331	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20		ovex 7482	. . . . 5 ⊢ (𝐼∈𝑔𝐽) ∈ V
21	3, 20	pm3.2i 476	. . . 4 ⊢ ({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V)
22		prv 36394	. . . 4 ⊢ (({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω)))
23	21, 22	mp1i 14	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω)))
24		goel 36313	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25		0ex 5265	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
26	25	snid 4623	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28		opelxpi 5693	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
29	27, 28	opelxpd 5695	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
30	24, 29	eqeltrd 2864	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ ({∅} × (ω × ω)))
31		fmla0xp 36349	. . . . . . . 8 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
32	30, 31	eleqtrrdi 2875	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
33	32	3adant3 1150	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
34		satefvfmla0 36384	. . . . . 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 599	. . . . 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 6918	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37		opex 5437	. . . . . . . . . . . . . 14 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
38	25, 37	op2nd 8042	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽⟩
39	36, 38	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
40	39	fveq2d 6918	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41		op1stg 8045	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
42	40, 41	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐼)
43	42	fveq2d 6918	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐼))
44	39	fveq2d 6918	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45		op2ndg 8046	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
46	44, 45	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐽)
47	46	fveq2d 6918	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐽))
48	43, 47	eleq12d 2858	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
49	48	rabbidv 3423	. . . . . . 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 1150	. . . . . 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 8905	. . . . . . . . . 10 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52		elirr 9629	. . . . . . . . . . . 12 ⊢ ¬ 𝑋 ∈ 𝑋
53		fvconst 7197	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎‘𝐼) = 𝑋)
54	53	3ad2antr1 1207	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐼) = 𝑋)
55		fvconst 7197	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎‘𝐽) = 𝑋)
56	55	3ad2antr2 1208	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐽) = 𝑋)
57	54, 56	eleq12d 2858	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → ((𝑎‘𝐼) ∈ (𝑎‘𝐽) ↔ 𝑋 ∈ 𝑋))
58	52, 57	mtbiri 330	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
59	58	ex 418	. . . . . . . . . 10 ⊢ (𝑎:ω⟶{𝑋} → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
60	51, 59	syl 18	. . . . . . . . 9 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
61	60	impcom 413	. . . . . . . 8 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) ∧ 𝑎 ∈ ({𝑋} ↑m ω)) → ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
62	61	ralrimiva 3158	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
63		rabeq0 4341	. . . . . . 7 ⊢ ({𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)} = ∅ ↔ ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
64	62, 63	sylibr 237	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → {𝑎 ∈ ({𝑋} ↑m ω) ∣ (𝑎‘𝐼) ∈ (𝑎‘𝐽)} = ∅)
65	50, 64	eqtrd 2798	. . . . 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 2798	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ∅)
67	66	eqeq1d 2765	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ({𝑋} ↑m ω) ↔ ∅ = ({𝑋} ↑m ω)))
68	23, 67	bitrd 282	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋}⊧(𝐼∈𝑔𝐽) ↔ ∅ = ({𝑋} ↑m ω)))
69	19, 68	mtbird 328	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ {𝑋}⊧(𝐼∈𝑔𝐽))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106   × cxp 5654  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449  ωcom 7909  1^st c1st 8031  2^nd c2nd 8032   ↑_m cmap 8883  ∈_𝑔cgoe 36299  Fmlacfmla 36303   Sat_∈ csate 36304  ⊧cprv 36305

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-reg 9621  ax-inf2 9677  ax-ac2 10578

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-2o 8513  df-er 8753  df-map 8885  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-card 10055  df-ac 10232  df-goel 36306  df-gona 36307  df-goal 36308  df-sat 36309  df-sate 36310  df-fmla 36311  df-prv 36312

This theorem is used by: (None)