Theorem scottexs 3543

Description: Theorem scheme version of scottex 3541. The collection of all x of minimum rank such that φ(x) is true, is a set.

Assertion

Ref	Expression
scottexs	⊢ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} ∈ V

Distinct variable group(s):   x,y   φ,y

Proof of Theorem scottexs

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (y ∈ {x∣φ} → ∀z y ∈ {x∣φ})
2		hbab1 1095	. . . 4 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
3		ax-17 925	. . . . 5 ⊢ ((rank ‘z) ⊆ (rank ‘y) → ∀x(rank ‘z) ⊆ (rank ‘y))
4	2, 3	hbral 1236	. . . 4 ⊢ (∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y) → ∀x∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y))
5		ax-17 925	. . . 4 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) → ∀z∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))
6		fveq2 2832	. . . . . 6 ⊢ (z = x → (rank ‘z) = (rank ‘x))
7	6	sseq1d 1527	. . . . 5 ⊢ (z = x → ((rank ‘z) ⊆ (rank ‘y) ↔ (rank ‘x) ⊆ (rank ‘y)))
8	7	biraldv 1219	. . . 4 ⊢ (z = x → (∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y) ↔ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)))
9	1, 2, 4, 5, 8	cbvrab 1425	. . 3 ⊢ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = {x ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)}
10		df-rab 1208	. . 3 ⊢ {x ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)} = {x∣(x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))}
11		abid 1094	. . . . 5 ⊢ (x ∈ {x∣φ} ↔ φ)
12		df-ral 1205	. . . . . 6 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y(y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)))
13		df-clab 1093	. . . . . . . 8 ⊢ (y ∈ {x∣φ} ↔ [y / x]φ)
14	13	imbi1i 161	. . . . . . 7 ⊢ ((y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)) ↔ ([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
15	14	bial 695	. . . . . 6 ⊢ (∀y(y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
16	12, 15	bitr 151	. . . . 5 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
17	11, 16	anbi12i 369	. . . 4 ⊢ ((x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)) ↔ (φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y))))
18	17	biabi 1181	. . 3 ⊢ {x∣(x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))} = {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))}
19	9, 10, 18	3eqtr 1123	. 2 ⊢ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))}
20		scottex 3541	. 2 ⊢ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} ∈ V
21	19, 20	eqeltrr 1160	1 ⊢ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} ∈ V

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = weq 797  [wsb 852  {cab 1090   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348   ⊆ wss 1487   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  hta 3619

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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