Theorem scott0 9776

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 3409	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2		rab0 4336	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
3	1, 2	eqtrdi 2782	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4		n0 4303	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		nfre1 3257	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)
6		eqid 2731	. . . . . . . . . 10 ⊢ (rank‘𝑥) = (rank‘𝑥)
7		rspe 3222	. . . . . . . . . 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 2220	. . . . . . . 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 6835	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ V
12		eqeq1 2735	. . . . . . . . . . . 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 3561	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
15	14	eximi 1836	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
16		excom 2165	. . . . . . . . 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 3057	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19		df-rex 3057	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
20	19	exbii 1849	. . . . . . . 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 4335	. . . . . 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 4983	. . . . . 6 ⊢ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)}
26		rankon 9685	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
27		eleq1 2819	. . . . . . . . . 10 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
28	26, 27	mpbiri 258	. . . . . . . . 9 ⊢ (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
29	28	rexlimivw 3129	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
30	29	abssi 4020	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31		onint 7723	. . . . . . 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 2835	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
34		nfii1 4979	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
35	34	nfeq2 2912	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
36		eqeq1 2735	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
37	35, 36	rexbid 3246	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
38	37	elabg 3632	. . . . . 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 3957	. . . . . . . . . 10 ⊢ (rank‘𝑦) ⊆ (rank‘𝑦)
41		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
42	41	sseq1d 3966	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
43	42	rspcev 3577	. . . . . . . . . 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 5005	. . . . . . . . 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 3960	. . . . . . . 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 3123	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
50	49	reximi 3070	. . . . 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 4339	. . . 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 2963	. 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 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395   ⊆ wss 3902  ∅c0 4283  ∩ cint 4897  ∩ ciin 4942  Oncon0 6306  ‘cfv 6481  rankcrnk 9653

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-r1 9654  df-rank 9655

This theorem is referenced by:  scott0s  9778  cplem1  9779  karden  9785  scott0f  38208  scotteld  44278