Theorem kardex 4725

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 1652	. . 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 1813	. . . . . 6 |- x e. V
3		breq1 2622	. . . . . 6 |- (z = x -> (z ~~ A <-> x ~~ A))
4	2, 3	elab 1897	. . . . 5 |- (x e. {z | z ~~ A} <-> x ~~ A)
5		df-ral 1649	. . . . . 6 |- (A.y e. {z | z ~~ A} (rank` x) (_ (rank` y) <-> A.y(y e. {z | z ~~ A} -> (rank` x) (_ (rank` y)))
6		visset 1813	. . . . . . . . 9 |- y e. V
7		breq1 2622	. . . . . . . . 9 |- (z = y -> (z ~~ A <-> y ~~ A))
8	6, 7	elab 1897	. . . . . . . 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 999	. . . . . 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 1575	. . 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 1495	. 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 4716	. 2 |- {x e. {z | z ~~ A} | A.y e. {z | z ~~ A} (rank` x) (_ (rank` y)} e. V
16	14, 15	eqeltrr 1545	1 |- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  {cab 1463  A.wral 1645  {crab 1648  Vcvv 1811   (_ wss 2047   class class class wbr 2619  ` cfv 3182   ~~ cen 4364  rankcrnk 4642

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-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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