Theorem kardex 4712

Description: The collection of all sets equinumerous to a set A and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222.

Assertion

Ref	Expression
kardex	|- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V

Distinct variable group:   x,y,A

Proof of Theorem kardex

Step	Hyp	Ref	Expression
1		df-rab 1651	. . 3 |- {x e. {z | z ~~ A} | A.y e. {z | z ~~ A} (rank` x) (_ (rank` y)} = {x | (x e. {z | z ~~ A} /\ A.y e. {z | z ~~ A} (rank` x) (_ (rank` y))}
2		visset 1811	. . . . . 6 |- x e. V
3		breq1 2619	. . . . . 6 |- (z = x -> (z ~~ A <-> x ~~ A))
4	2, 3	elab 1895	. . . . 5 |- (x e. {z | z ~~ A} <-> x ~~ A)
5		df-ral 1648	. . . . . 6 |- (A.y e. {z | z ~~ A} (rank` x) (_ (rank` y) <-> A.y(y e. {z | z ~~ A} -> (rank` x) (_ (rank` y)))
6		visset 1811	. . . . . . . . 9 |- y e. V
7		breq1 2619	. . . . . . . . 9 |- (z = y -> (z ~~ A <-> y ~~ A))
8	6, 7	elab 1895	. . . . . . . 8 |- (y e. {z | z ~~ A} <-> y ~~ A)
9	8	imbi1i 186	. . . . . . 7 |- ((y e. {z | z ~~ A} -> (rank` x) (_ (rank` y)) <-> (y ~~ A -> (rank` x) (_ (rank` y)))
10	9	albii 998	. . . . . 6 |- (A.y(y e. {z | z ~~ A} -> (rank` x) (_ (rank` y)) <-> A.y(y ~~ A -> (rank` x) (_ (rank` y)))
11	5, 10	bitr 173	. . . . 5 |- (A.y e. {z | z ~~ A} (rank` x) (_ (rank` y) <-> A.y(y ~~ A -> (rank` x) (_ (rank` y)))
12	4, 11	anbi12i 482	. . . 4 |- ((x e. {z | z ~~ A} /\ A.y e. {z | z ~~ A} (rank` x) (_ (rank` y)) <-> (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y))))
13	12	abbii 1574	. . 3 |- {x | (x e. {z | z ~~ A} /\ A.y e. {z | z ~~ A} (rank` x) (_ (rank` y))} = {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))}
14	1, 13	eqtr 1494	. 2 |- {x e. {z | z ~~ A} | A.y e. {z | z ~~ A} (rank` x) (_ (rank` y)} = {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))}
15		scottex 4703	. 2 |- {x e. {z | z ~~ A} | A.y e. {z | z ~~ A} (rank` x) (_ (rank` y)} e. V
16	14, 15	eqeltrr 1544	1 |- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   e. wcel 957  {cab 1463  A.wral 1644  {crab 1647  Vcvv 1809   (_ wss 2045   class class class wbr 2616  ` cfv 3179   ~~ cen 4361  rankcrnk 4629

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 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-reg 4580  ax-inf2 4612

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 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-sbc 1940  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195  df-rdg 3929  df-r1 4630  df-rank 4631

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