Theorem scott0 9881

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 3447	. . 3 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2		rab0 4383	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
3	1, 2	eqtrdi 2789	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4		n0 4347	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		nfre1 3283	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)
6		eqid 2733	. . . . . . . . . 10 ⊢ (rank‘𝑥) = (rank‘𝑥)
7		rspe 3247	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
8	6, 7	mpan2 690	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥))
9	5, 8	exlimi 2211	. . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ V
12		eqeq1 2737	. . . . . . . . . . . 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 3597	. . . . . . . . . 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 2163	. . . . . . . . 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 3072	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19		df-rex 3072	. . . . . . . . 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 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 4381	. . . . . 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 5038	. . . . . 6 ⊢ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)}
26		rankon 9790	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
27		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
28	26, 27	mpbiri 258	. . . . . . . . 9 ⊢ (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
29	28	rexlimivw 3152	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
30	29	abssi 4068	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31		onint 7778	. . . . . . 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 689	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
33	25, 32	eqeltrid 2838	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥)})
34		nfii1 5033	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
35	34	nfeq2 2921	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥)
36		eqeq1 2737	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
37	35, 36	rexbid 3272	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∃𝑥 ∈ 𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥)))
38	37	elabg 3667	. . . . . 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 4005	. . . . . . . . . 10 ⊢ (rank‘𝑦) ⊆ (rank‘𝑦)
41		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
42	41	sseq1d 4014	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
43	42	rspcev 3613	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
44	40, 43	mpan2 690	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45		iinss 5060	. . . . . . . . 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 4008	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
48	46, 47	imbitrid 243	. . . . . . 7 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
49	48	ralrimiv 3146	. . . . . 6 ⊢ (∩ 𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
50	49	reximi 3085	. . . . 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 4386	. . . 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 2977	. 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 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433   ⊆ wss 3949  ∅c0 4323  ∩ cint 4951  ∩ ciin 4999  Oncon0 6365  ‘cfv 6544  rankcrnk 9758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760

This theorem is referenced by:  scott0s  9883  cplem1  9884  karden  9890  scott0f  37037  scotteld  43005