Theorem scott0 9575

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 3408	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2		rab0 4313	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
3	1, 2	eqtrdi 2795	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4		n0 4277	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		nfre1 3234	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)
6		eqid 2738	. . . . . . . . . 10 ⊢ (rank‘𝑥) = (rank‘𝑥)
7		rspe 3232	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
8	6, 7	mpan2 687	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
9	5, 8	exlimi 2213	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
10	4, 9	sylbi 216	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
11		fvex 6769	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ V
12		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
13	12	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑦 = (rank‘𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
14	11, 13	spcev 3535	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
15	14	eximi 1838	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
16		excom 2164	. . . . . . . . 9 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
17	15, 16	sylibr 233	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
18		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19		df-rex 3069	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
20	19	exbii 1851	. . . . . . . 8 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = (rank‘𝑥)))
21	17, 18, 20	3imtr4i 291	. . . . . . 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 4311	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥))
24	22, 23	sylibr 233	. . . . 5 ⊢ (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
25	11	dfiin2 4960	. . . . . 6 ⊢ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)}
26		rankon 9484	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
27		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
28	26, 27	mpbiri 257	. . . . . . . . 9 ⊢ (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
29	28	rexlimivw 3210	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
30	29	abssi 3999	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31		onint 7617	. . . . . . 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 686	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
33	25, 32	eqeltrid 2843	. . . . 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 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
36		eqeq1 2742	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
37	35, 36	rexbid 3248	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
38	37	elabg 3600	. . . . . 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 266	. . . . 5 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
40		ssid 3939	. . . . . . . . . 10 ⊢ (rank‘𝑦) ⊆ (rank‘𝑦)
41		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
42	41	sseq1d 3948	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
43	42	rspcev 3552	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
44	40, 43	mpan2 687	. . . . . . . . 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 3942	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
48	46, 47	syl5ib 243	. . . . . . 7 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
49	48	ralrimiv 3106	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
50	49	reximi 3174	. . . . 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 4316	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
53	51, 52	sylibr 233	. . 3 ⊢ (𝐴 ≠ ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
54	53	necon4i 2978	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
55	3, 54	impbii 208	1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876  ∩ ciin 4922  Oncon0 6251  ‘cfv 6418  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  scott0s  9577  cplem1  9578  karden  9584  scott0f  36254  scotteld  41753