Theorem scott0 9839

Description: Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
scott0	⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem scott0

Step	Hyp	Ref	Expression
1		rabeq 3420	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2		rab0 4349	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
3	1, 2	eqtrdi 2780	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4		n0 4316	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		nfre1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)
6		eqid 2729	. . . . . . . . . 10 ⊢ (rank‘𝑥) = (rank‘𝑥)
7		rspe 3227	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
8	6, 7	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
9	5, 8	exlimi 2218	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
10	4, 9	sylbi 217	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
11		fvex 6871	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ V
12		eqeq1 2733	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
13	12	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑦 = (rank‘𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
14	11, 13	spcev 3572	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
15	14	eximi 1835	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
16		excom 2163	. . . . . . . . 9 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
17	15, 16	sylibr 234	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
18		df-rex 3054	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19		df-rex 3054	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
20	19	exbii 1848	. . . . . . . 8 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
21	17, 18, 20	3imtr4i 292	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥))
22	10, 21	syl 17	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥))
23		abn0 4348	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥))
24	22, 23	sylibr 234	. . . . 5 ⊢ (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
25	11	dfiin2 4998	. . . . . 6 ⊢ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)}
26		rankon 9748	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
27		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
28	26, 27	mpbiri 258	. . . . . . . . 9 ⊢ (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
29	28	rexlimivw 3130	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
30	29	abssi 4033	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31		onint 7766	. . . . . . 7 ⊢ (({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
32	30, 31	mpan 690	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
33	25, 32	eqeltrid 2832	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
34		nfii1 4993	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
35	34	nfeq2 2909	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
36		eqeq1 2733	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
37	35, 36	rexbid 3251	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
38	37	elabg 3643	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} → (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
39	38	ibi 267	. . . . 5 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
40		ssid 3969	. . . . . . . . . 10 ⊢ (rank‘𝑦) ⊆ (rank‘𝑦)
41		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
42	41	sseq1d 3978	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
43	42	rspcev 3588	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
44	40, 43	mpan2 691	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45		iinss 5020	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
46	44, 45	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47		sseq1 3972	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
48	46, 47	imbitrid 244	. . . . . . 7 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
49	48	ralrimiv 3124	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
50	49	reximi 3067	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
51	24, 33, 39, 50	4syl 19	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
52		rabn0 4352	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
53	51, 52	sylibr 234	. . 3 ⊢ (𝐴 ≠ ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
54	53	necon4i 2960	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
55	3, 54	impbii 209	1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405   ⊆ wss 3914  ∅c0 4296  ∩ cint 4910  ∩ ciin 4956  Oncon0 6332  ‘cfv 6511  rankcrnk 9716

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-ral 3045  df-rex 3054  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-iin 4958  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-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-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718

This theorem is referenced by:  scott0s  9841  cplem1  9842  karden  9848  scott0f  38163  scotteld  44235