Theorem en3d 3304

Description: Equinumerosity inference from an implicit one-to-one onto function.

Hypotheses

Ref	Expression
en3d.1	|- (ph -> A e. V)
en3d.2	|- (ph -> (x e. A -> C e. B))
en3d.3	|- (ph -> (y e. B -> D e. A))
en3d.4	|- (ph -> ((x e. A /\ y e. B) -> (x = D <-> y = C)))

Assertion

Ref	Expression
en3d	|- (ph -> A ~~ B)

Distinct variable group(s):   x,y,A   x,B,y   y,C   x,D   ph,x,y

Proof of Theorem en3d

Step	Hyp	Ref	Expression
1		en3d.1	. 2 |- (ph -> A e. V)
2		en3d.2	. . 3 |- (ph -> (x e. A -> C e. B))
3		elisset 1354	. . 3 |- (C e. B -> C e. V)
4	2, 3	syl6 23	. 2 |- (ph -> (x e. A -> C e. V))
5		en3d.3	. . 3 |- (ph -> (y e. B -> D e. A))
6		elisset 1354	. . 3 |- (D e. A -> D e. V)
7	5, 6	syl6 23	. 2 |- (ph -> (y e. B -> D e. V))
8		eleq1a 1158	. . . . . . 7 |- (C e. B -> (y = C -> y e. B))
9	2, 8	syl6 23	. . . . . 6 |- (ph -> (x e. A -> (y = C -> y e. B)))
10	9	imp32 281	. . . . 5 |- ((ph /\ (x e. A /\ y = C)) -> y e. B)
11		en3d.4	. . . . . . . . . 10 |- (ph -> ((x e. A /\ y e. B) -> (x = D <-> y = C)))
12	11	imp 277	. . . . . . . . 9 |- ((ph /\ (x e. A /\ y e. B)) -> (x = D <-> y = C))
13	12	biimpar 325	. . . . . . . 8 |- (((ph /\ (x e. A /\ y e. B)) /\ y = C) -> x = D)
14	13	exp42 300	. . . . . . 7 |- (ph -> (x e. A -> (y e. B -> (y = C -> x = D))))
15	14	com34 36	. . . . . 6 |- (ph -> (x e. A -> (y = C -> (y e. B -> x = D))))
16	15	imp32 281	. . . . 5 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B -> x = D))
17	10, 16	jcai 237	. . . 4 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B /\ x = D))
18	17	exp 291	. . 3 |- (ph -> ((x e. A /\ y = C) -> (y e. B /\ x = D)))
19		eleq1a 1158	. . . . . . 7 |- (D e. A -> (x = D -> x e. A))
20	5, 19	syl6 23	. . . . . 6 |- (ph -> (y e. B -> (x = D -> x e. A)))
21	20	imp32 281	. . . . 5 |- ((ph /\ (y e. B /\ x = D)) -> x e. A)
22	12	biimpa 324	. . . . . . . . 9 |- (((ph /\ (x e. A /\ y e. B)) /\ x = D) -> y = C)
23	22	exp42 300	. . . . . . . 8 |- (ph -> (x e. A -> (y e. B -> (x = D -> y = C))))
24	23	com23 32	. . . . . . 7 |- (ph -> (y e. B -> (x e. A -> (x = D -> y = C))))
25	24	com34 36	. . . . . 6 |- (ph -> (y e. B -> (x = D -> (x e. A -> y = C))))
26	25	imp32 281	. . . . 5 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A -> y = C))
27	21, 26	jcai 237	. . . 4 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A /\ y = C))
28	27	exp 291	. . 3 |- (ph -> ((y e. B /\ x = D) -> (x e. A /\ y = C)))
29	18, 28	impbid 397	. 2 |- (ph -> ((x e. A /\ y = C) <-> (y e. B /\ x = D)))
30	1, 4, 7, 29	en2d 3303	1 |- (ph -> A ~~ B)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054   ~~ cen 3271

This theorem is referenced by:  en3 3306  fundmen 3333  mapunen 3397  ssenen 3399

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-fo 2436  df-f1o 2437  df-en 3274

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