Theorem scott0 9317

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 3485	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2		rab0 4339	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
3	1, 2	syl6eq 2874	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4		n0 4312	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		nfre1 3308	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)
6		eqid 2823	. . . . . . . . . 10 ⊢ (rank‘𝑥) = (rank‘𝑥)
7		rspe 3306	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
8	6, 7	mpan2 689	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
9	5, 8	exlimi 2217	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
10	4, 9	sylbi 219	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
11		fvex 6685	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ V
12		eqeq1 2827	. . . . . . . . . . . 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 3609	. . . . . . . . . 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 2169	. . . . . . . . 9 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
17	15, 16	sylibr 236	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
18		df-rex 3146	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19		df-rex 3146	. . . . . . . . 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 294	. . . . . . 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 4338	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥))
24	22, 23	sylibr 236	. . . . 5 ⊢ (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
25	11	dfiin2 4961	. . . . . 6 ⊢ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)}
26		rankon 9226	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
27		eleq1 2902	. . . . . . . . . 10 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
28	26, 27	mpbiri 260	. . . . . . . . 9 ⊢ (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
29	28	rexlimivw 3284	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
30	29	abssi 4048	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31		onint 7512	. . . . . . 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 688	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
33	25, 32	eqeltrid 2919	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
34		nfii1 4956	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
35	34	nfeq2 2997	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
36		eqeq1 2827	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
37	35, 36	rexbid 3322	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
38	37	elabg 3668	. . . . . 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 269	. . . . 5 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
40		ssid 3991	. . . . . . . . . 10 ⊢ (rank‘𝑦) ⊆ (rank‘𝑦)
41		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
42	41	sseq1d 4000	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
43	42	rspcev 3625	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
44	40, 43	mpan2 689	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45		iinss 4982	. . . . . . . . 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 3994	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
48	46, 47	syl5ib 246	. . . . . . 7 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
49	48	ralrimiv 3183	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
50	49	reximi 3245	. . . . 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 4341	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
53	51, 52	sylibr 236	. . 3 ⊢ (𝐴 ≠ ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
54	53	necon4i 3053	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
55	3, 54	impbii 211	1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144   ⊆ wss 3938  ∅c0 4293  ∩ cint 4878  ∩ ciin 4922  Oncon0 6193  ‘cfv 6357  rankcrnk 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195  df-rank 9196

This theorem is referenced by:  scott0s  9319  cplem1  9320  karden  9326  scott0f  35449  scotteld  40589