Theorem scottex 3541

Description: Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set.

Assertion

Ref	Expression
scottex	⊢ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V

Distinct variable group(s):   x,y,A

Proof of Theorem scottex

Step	Hyp	Ref	Expression
1		0ex 1745	. . . 4 ⊢ ∅ ∈ V
2		eleq1 1149	. . . 4 ⊢ (A = ∅ → (A ∈ V ↔ ∅ ∈ V))
3	1, 2	mpbiri 169	. . 3 ⊢ (A = ∅ → A ∈ V)
4		rabexg 1705	. . 3 ⊢ (A ∈ V → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
5	3, 4	syl 12	. 2 ⊢ (A = ∅ → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
6		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃y y ∈ A)
7		hbra1 1237	. . . . . 6 ⊢ (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → ∀y∀y ∈ A (rank ‘x) ⊆ (rank ‘y))
8		ax-17 925	. . . . . 6 ⊢ (z ∈ A → ∀y z ∈ A)
9	7, 8	hbrab 1311	. . . . 5 ⊢ (z ∈ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} → ∀y z ∈ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)})
10		ax-17 925	. . . . 5 ⊢ (z ∈ V → ∀y z ∈ V)
11	9, 10	hbel 1172	. . . 4 ⊢ ({x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V → ∀y{x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
12		ra4 1243	. . . . . . . . 9 ⊢ (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → (y ∈ A → (rank ‘x) ⊆ (rank ‘y)))
13	12	com12 13	. . . . . . . 8 ⊢ (y ∈ A → (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → (rank ‘x) ⊆ (rank ‘y)))
14	13	a1d 14	. . . . . . 7 ⊢ (y ∈ A → (x ∈ A → (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → (rank ‘x) ⊆ (rank ‘y))))
15	14	r19.21aiv 1259	. . . . . 6 ⊢ (y ∈ A → ∀x ∈ A (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → (rank ‘x) ⊆ (rank ‘y)))
16		ss2rab 1553	. . . . . 6 ⊢ ({x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ⊆ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} ↔ ∀x ∈ A (∀y ∈ A (rank ‘x) ⊆ (rank ‘y) → (rank ‘x) ⊆ (rank ‘y)))
17	15, 16	sylibr 175	. . . . 5 ⊢ (y ∈ A → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ⊆ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)})
18		rankon 3515	. . . . . . . 8 ⊢ (rank ‘y) ∈ On
19		fveq2 2832	. . . . . . . . . . . 12 ⊢ (x = w → (rank ‘x) = (rank ‘w))
20	19	sseq1d 1527	. . . . . . . . . . 11 ⊢ (x = w → ((rank ‘x) ⊆ (rank ‘y) ↔ (rank ‘w) ⊆ (rank ‘y)))
21	20	elrab 1422	. . . . . . . . . 10 ⊢ (w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} ↔ (w ∈ A ∧ (rank ‘w) ⊆ (rank ‘y)))
22	21	pm3.27bd 263	. . . . . . . . 9 ⊢ (w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} → (rank ‘w) ⊆ (rank ‘y))
23	22	rgen 1247	. . . . . . . 8 ⊢ ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ (rank ‘y)
24		sseq2 1522	. . . . . . . . . 10 ⊢ (z = (rank ‘y) → ((rank ‘w) ⊆ z ↔ (rank ‘w) ⊆ (rank ‘y)))
25	24	biraldv 1219	. . . . . . . . 9 ⊢ (z = (rank ‘y) → (∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ z ↔ ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ (rank ‘y)))
26	25	rcla4ev 1403	. . . . . . . 8 ⊢ (((rank ‘y) ∈ On ∧ ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ (rank ‘y)) → ∃z ∈ On ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ z)
27	18, 23, 26	mp2an 520	. . . . . . 7 ⊢ ∃z ∈ On ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ z
28		bndrank 3526	. . . . . . 7 ⊢ (∃z ∈ On ∀w ∈ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} (rank ‘w) ⊆ z → {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} ∈ V)
29	27, 28	ax-mp 6	. . . . . 6 ⊢ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} ∈ V
30	29	ssex 1700	. . . . 5 ⊢ ({x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ⊆ {x ∈ A∣(rank ‘x) ⊆ (rank ‘y)} → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
31	17, 30	syl 12	. . . 4 ⊢ (y ∈ A → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
32	11, 31	19.23ai 746	. . 3 ⊢ (∃y y ∈ A → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
33	6, 32	sylbi 174	. 2 ⊢ (¬ A = ∅ → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V)
34	5, 33	pm2.61i 110	1 ⊢ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} ∈ V

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  Oncon0 2199   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  scottexs 3543  cplem2 3546  kardex 3550

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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