Theorem en3d 4391

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 groups:   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 1814	. . 3 |- (C e. B -> C e. V)
4	2, 3	syl6 22	. 2 |- (ph -> (x e. A -> C e. V))
5		en3d.3	. . 3 |- (ph -> (y e. B -> D e. A))
6		elisset 1814	. . 3 |- (D e. A -> D e. V)
7	5, 6	syl6 22	. 2 |- (ph -> (y e. B -> D e. V))
8		eleq1a 1541	. . . . . 6 |- (C e. B -> (y = C -> y e. B))
9	2, 8	syl6 22	. . . . 5 |- (ph -> (x e. A -> (y = C -> y e. B)))
10	9	imp32 363	. . . 4 |- ((ph /\ (x e. A /\ y = C)) -> y e. B)
11		en3d.4	. . . . . . . . 9 |- (ph -> ((x e. A /\ y e. B) -> (x = D <-> y = C)))
12	11	imp 350	. . . . . . . 8 |- ((ph /\ (x e. A /\ y e. B)) -> (x = D <-> y = C))
13	12	biimpar 417	. . . . . . 7 |- (((ph /\ (x e. A /\ y e. B)) /\ y = C) -> x = D)
14	13	exp42 383	. . . . . 6 |- (ph -> (x e. A -> (y e. B -> (y = C -> x = D))))
15	14	com34 36	. . . . 5 |- (ph -> (x e. A -> (y = C -> (y e. B -> x = D))))
16	15	imp32 363	. . . 4 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B -> x = D))
17	10, 16	jcai 289	. . 3 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B /\ x = D))
18		eleq1a 1541	. . . . . 6 |- (D e. A -> (x = D -> x e. A))
19	5, 18	syl6 22	. . . . 5 |- (ph -> (y e. B -> (x = D -> x e. A)))
20	19	imp32 363	. . . 4 |- ((ph /\ (y e. B /\ x = D)) -> x e. A)
21	12	biimpa 416	. . . . . . . 8 |- (((ph /\ (x e. A /\ y e. B)) /\ x = D) -> y = C)
22	21	exp42 383	. . . . . . 7 |- (ph -> (x e. A -> (y e. B -> (x = D -> y = C))))
23	22	com23 32	. . . . . 6 |- (ph -> (y e. B -> (x e. A -> (x = D -> y = C))))
24	23	com34 36	. . . . 5 |- (ph -> (y e. B -> (x = D -> (x e. A -> y = C))))
25	24	imp32 363	. . . 4 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A -> y = C))
26	20, 25	jcai 289	. . 3 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A /\ y = C))
27	17, 26	impbida 518	. 2 |- (ph -> ((x e. A /\ y = C) <-> (y e. B /\ x = D)))
28	1, 4, 7, 27	en2d 4390	1 |- (ph -> A ~~ B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808   class class class wbr 2615   ~~ cen 4357

This theorem is referenced by:  en3 4393  fundmen 4418  mapunen 4491  ssenen 4493

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-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  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-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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