Theorem scott0s 3544

Description: Theorem scheme version of scott0 3542. The collection of all x of minimum rank such that φ(x) is true, is not empty iff there is an x such that φ(x) holds.

Assertion

Ref	Expression
scott0s	⊢ (∃xφ ↔ ¬ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} = ∅)

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

Proof of Theorem scott0s

Step	Hyp	Ref	Expression
1		abn0 1715	. 2 ⊢ (¬ {x∣φ} = ∅ ↔ ∃xφ)
2		scott0 3542	. . . 4 ⊢ ({x∣φ} = ∅ ↔ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = ∅)
3		ax-17 925	. . . . . . 7 ⊢ (y ∈ {x∣φ} → ∀z y ∈ {x∣φ})
4		hbab1 1095	. . . . . . 7 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
5		ax-17 925	. . . . . . . 8 ⊢ ((rank ‘z) ⊆ (rank ‘y) → ∀x(rank ‘z) ⊆ (rank ‘y))
6	4, 5	hbral 1236	. . . . . . 7 ⊢ (∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y) → ∀x∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y))
7		ax-17 925	. . . . . . 7 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) → ∀z∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))
8		fveq2 2832	. . . . . . . . 9 ⊢ (z = x → (rank ‘z) = (rank ‘x))
9	8	sseq1d 1527	. . . . . . . 8 ⊢ (z = x → ((rank ‘z) ⊆ (rank ‘y) ↔ (rank ‘x) ⊆ (rank ‘y)))
10	9	biraldv 1219	. . . . . . 7 ⊢ (z = x → (∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y) ↔ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)))
11	3, 4, 6, 7, 10	cbvrab 1425	. . . . . 6 ⊢ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = {x ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)}
12		df-rab 1208	. . . . . 6 ⊢ {x ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)} = {x∣(x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))}
13		abid 1094	. . . . . . . 8 ⊢ (x ∈ {x∣φ} ↔ φ)
14		df-ral 1205	. . . . . . . . 9 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y(y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)))
15		df-clab 1093	. . . . . . . . . . 11 ⊢ (y ∈ {x∣φ} ↔ [y / x]φ)
16	15	imbi1i 161	. . . . . . . . . 10 ⊢ ((y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)) ↔ ([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
17	16	bial 695	. . . . . . . . 9 ⊢ (∀y(y ∈ {x∣φ} → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
18	14, 17	bitr 151	. . . . . . . 8 ⊢ (∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))
19	13, 18	anbi12i 369	. . . . . . 7 ⊢ ((x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y)) ↔ (φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y))))
20	19	biabi 1181	. . . . . 6 ⊢ {x∣(x ∈ {x∣φ} ∧ ∀y ∈ {x∣φ} (rank ‘x) ⊆ (rank ‘y))} = {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))}
21	11, 12, 20	3eqtr 1123	. . . . 5 ⊢ {z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))}
22	21	cleq1i 1108	. . . 4 ⊢ ({z ∈ {x∣φ}∣∀y ∈ {x∣φ} (rank ‘z) ⊆ (rank ‘y)} = ∅ ↔ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} = ∅)
23	2, 22	bitr 151	. . 3 ⊢ ({x∣φ} = ∅ ↔ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} = ∅)
24	23	negbii 162	. 2 ⊢ (¬ {x∣φ} = ∅ ↔ ¬ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} = ∅)
25	1, 24	bitr3 153	1 ⊢ (∃xφ ↔ ¬ {x∣(φ ∧ ∀y([y / x]φ → (rank ‘x) ⊆ (rank ‘y)))} = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487  ∅c0 1707   ‘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-iin 1997  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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