Theorem prv1n 35644

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 2737	. . . . . 6 ⊢ (ω × {𝑋}) = (ω × {𝑋})
2		omex 9564	. . . . . . . 8 ⊢ ω ∈ V
3		snex 5385	. . . . . . . 8 ⊢ {𝑋} ∈ V
4	2, 3	xpex 7708	. . . . . . 7 ⊢ (ω × {𝑋}) ∈ V
5		eqeq1 2741	. . . . . . 7 ⊢ (𝑎 = (ω × {𝑋}) → (𝑎 = (ω × {𝑋}) ↔ (ω × {𝑋}) = (ω × {𝑋})))
6	4, 5	spcev 3562	. . . . . 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 8788	. . . . . . . 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 7159	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑎:ω⟶{𝑋} ↔ 𝑎 = (ω × {𝑋})))
12	11	3ad2ant3 1136	. . . . . . 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 1923	. . . . 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 4306	. . . 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 2744	. . 3 ⊢ (({𝑋} ↑m ω) = ∅ ↔ ∅ = ({𝑋} ↑m ω))
19	17, 18	sylnib 328	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ ∅ = ({𝑋} ↑m ω))
20		ovex 7401	. . . . 5 ⊢ (𝐼∈𝑔𝐽) ∈ V
21	3, 20	pm3.2i 470	. . . 4 ⊢ ({𝑋} ∈ V ∧ (𝐼∈𝑔𝐽) ∈ V)
22		prv 35641	. . . 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 35560	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
25		0ex 5254	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
26	25	snid 4621	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ {∅})
28		opelxpi 5669	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
29	27, 28	opelxpd 5671	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ ({∅} × (ω × ω)))
30	24, 29	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ ({∅} × (ω × ω)))
31		fmla0xp 35596	. . . . . . . 8 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
32	30, 31	eleqtrrdi 2848	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
33	32	3adant3 1133	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → (𝐼∈𝑔𝐽) ∈ (Fmla‘∅))
34		satefvfmla0 35631	. . . . . 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 588	. . . . 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 6846	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩))
37		opex 5419	. . . . . . . . . . . . . 14 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
38	25, 37	op2nd 7952	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨∅, ⟨𝐼, 𝐽⟩⟩) = ⟨𝐼, 𝐽⟩
39	36, 38	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(𝐼∈𝑔𝐽)) = ⟨𝐼, 𝐽⟩)
40	39	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = (1st ‘⟨𝐼, 𝐽⟩))
41		op1stg 7955	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘⟨𝐼, 𝐽⟩) = 𝐼)
42	40, 41	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (1st ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐼)
43	42	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐼))
44	39	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = (2nd ‘⟨𝐼, 𝐽⟩))
45		op2ndg 7956	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘⟨𝐼, 𝐽⟩) = 𝐽)
46	44, 45	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (2nd ‘(2nd ‘(𝐼∈𝑔𝐽))) = 𝐽)
47	46	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) = (𝑎‘𝐽))
48	43, 47	eleq12d 2831	. . . . . . . 8 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘(1st ‘(2nd ‘(𝐼∈𝑔𝐽)))) ∈ (𝑎‘(2nd ‘(2nd ‘(𝐼∈𝑔𝐽)))) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
49	48	rabbidv 3408	. . . . . . 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 1133	. . . . . 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 8798	. . . . . . . . . 10 ⊢ (𝑎 ∈ ({𝑋} ↑m ω) → 𝑎:ω⟶{𝑋})
52		elirr 9516	. . . . . . . . . . . 12 ⊢ ¬ 𝑋 ∈ 𝑋
53		fvconst 7118	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐼 ∈ ω) → (𝑎‘𝐼) = 𝑋)
54	53	3ad2antr1 1190	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐼) = 𝑋)
55		fvconst 7118	. . . . . . . . . . . . . 14 ⊢ ((𝑎:ω⟶{𝑋} ∧ 𝐽 ∈ ω) → (𝑎‘𝐽) = 𝑋)
56	55	3ad2antr2 1191	. . . . . . . . . . . . 13 ⊢ ((𝑎:ω⟶{𝑋} ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉)) → (𝑎‘𝐽) = 𝑋)
57	54, 56	eleq12d 2831	. . . . . . . . . . . 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 3130	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ∀𝑎 ∈ ({𝑋} ↑m ω) ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
63		rabeq0 4342	. . . . . . 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 2772	. . . . 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 2772	. . . 4 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ({𝑋} Sat∈ (𝐼∈𝑔𝐽)) = ∅)
67	66	eqeq1d 2739	. . 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442  ∅c0 4287  {csn 4582  ⟨cop 4588   class class class wbr 5100   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ωcom 7818  1^st c1st 7941  2^nd c2nd 7942   ↑_m cmap 8775  ∈_𝑔cgoe 35546  Fmlacfmla 35550   Sat_∈ csate 35551  ⊧cprv 35552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-ac 10038  df-goel 35553  df-gona 35554  df-goal 35555  df-sat 35556  df-sate 35557  df-fmla 35558  df-prv 35559

This theorem is referenced by: (None)