Theorem scott0s 4702

Description: Theorem scheme version of scott0 4700. The collection of all x of minimum rank such that ph(x) is true, is not empty iff there is an x such that ph(x) holds.

Assertion

Ref	Expression
scott0s	|- (E.xph <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} =/= (/))

Distinct variable groups:   x,y   ph,y

Proof of Theorem scott0s

Step	Hyp	Ref	Expression
1		abn0 2287	. 2 |- ({x | ph} =/= (/) <-> E.xph)
2		scott0 4700	. . . 4 |- ({x | ph} = (/) <-> {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = (/))
3		ax-17 970	. . . . . . 7 |- (y e. {x | ph} -> A.z y e. {x | ph})
4		hbab1 1465	. . . . . . 7 |- (y e. {x | ph} -> A.x y e. {x | ph})
5		ax-17 970	. . . . . . . 8 |- ((rank` z) (_ (rank` y) -> A.x(rank` z) (_ (rank` y))
6	4, 5	hbral 1684	. . . . . . 7 |- (A.y e. {x | ph} (rank` z) (_ (rank` y) -> A.xA.y e. {x | ph} (rank` z) (_ (rank` y))
7		ax-17 970	. . . . . . 7 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) -> A.zA.y e. {x | ph} (rank` x) (_ (rank` y))
8		fveq2 3719	. . . . . . . . 9 |- (z = x -> (rank` z) = (rank` x))
9	8	sseq1d 2085	. . . . . . . 8 |- (z = x -> ((rank` z) (_ (rank` y) <-> (rank` x) (_ (rank` y)))
10	9	ralbidv 1661	. . . . . . 7 |- (z = x -> (A.y e. {x | ph} (rank` z) (_ (rank` y) <-> A.y e. {x | ph} (rank` x) (_ (rank` y)))
11	3, 4, 6, 7, 10	cbvrab 1907	. . . . . 6 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = {x e. {x | ph} | A.y e. {x | ph} (rank` x) (_ (rank` y)}
12		df-rab 1650	. . . . . 6 |- {x e. {x | ph} | A.y e. {x | ph} (rank` x) (_ (rank` y)} = {x | (x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y))}
13		abid 1464	. . . . . . . 8 |- (x e. {x | ph} <-> ph)
14		df-ral 1647	. . . . . . . . 9 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)))
15		df-clab 1463	. . . . . . . . . . 11 |- (y e. {x | ph} <-> [y / x]ph)
16	15	imbi1i 186	. . . . . . . . . 10 |- ((y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> ([y / x]ph -> (rank` x) (_ (rank` y)))
17	16	albii 998	. . . . . . . . 9 |- (A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
18	14, 17	bitr 173	. . . . . . . 8 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
19	13, 18	anbi12i 482	. . . . . . 7 |- ((x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y)) <-> (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y))))
20	19	abbii 1573	. . . . . 6 |- {x | (x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y))} = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}
21	11, 12, 20	3eqtr 1497	. . . . 5 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}
22	21	eqeq1i 1480	. . . 4 |- ({z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = (/) <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} = (/))
23	2, 22	bitr 173	. . 3 |- ({x | ph} = (/) <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} = (/))
24	23	necon3bii 1596	. 2 |- ({x | ph} =/= (/) <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} =/= (/))
25	1, 24	bitr3 175	1 |- (E.xph <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} =/= (/))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  [wsbc 1169  {cab 1462   =/= wne 1583  A.wral 1643  {crab 1646   (_ wss 2044  (/)c0 2277  ` cfv 3178  rankcrnk 4625

This theorem is referenced by:  hta 4711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-iin 2565  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-r1 4626  df-rank 4627

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