Theorem scott0s 4719

Description: Theorem scheme version of scott0 4717. 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 2290	. 2 |- ({x | ph} =/= (/) <-> E.xph)
2		scott0 4717	. . . 4 |- ({x | ph} = (/) <-> {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = (/))
3		ax-17 971	. . . . . . 7 |- (y e. {x | ph} -> A.z y e. {x | ph})
4		hbab1 1466	. . . . . . 7 |- (y e. {x | ph} -> A.x y e. {x | ph})
5		ax-17 971	. . . . . . . 8 |- ((rank` z) (_ (rank` y) -> A.x(rank` z) (_ (rank` y))
6	4, 5	hbral 1686	. . . . . . 7 |- (A.y e. {x | ph} (rank` z) (_ (rank` y) -> A.xA.y e. {x | ph} (rank` z) (_ (rank` y))
7		ax-17 971	. . . . . . 7 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) -> A.zA.y e. {x | ph} (rank` x) (_ (rank` y))
8		fveq2 3724	. . . . . . . . 9 |- (z = x -> (rank` z) = (rank` x))
9	8	sseq1d 2088	. . . . . . . 8 |- (z = x -> ((rank` z) (_ (rank` y) <-> (rank` x) (_ (rank` y)))
10	9	ralbidv 1663	. . . . . . 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 1910	. . . . . 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 1652	. . . . . 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 1465	. . . . . . . 8 |- (x e. {x | ph} <-> ph)
14		df-ral 1649	. . . . . . . . 9 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)))
15		df-clab 1464	. . . . . . . . . . 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 999	. . . . . . . . 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 1575	. . . . . 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 1499	. . . . 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 1482	. . . 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 1598	. 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 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  {cab 1463   =/= wne 1585  A.wral 1645  {crab 1648   (_ wss 2047  (/)c0 2280  ` cfv 3182  rankcrnk 4642

This theorem is referenced by:  hta 4728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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