Theorem cdaen 6073

Description: Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.

Hypotheses

Ref	Expression
cdaen.1	|- A e. _V
cdaen.2	|- B e. _V
cdaen.3	|- C e. _V
cdaen.4	|- D e. _V

Assertion

Ref	Expression
cdaen	|- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))

Proof of Theorem cdaen

Step	Hyp	Ref	Expression
1		xp01disj 5188	. . . 4 |- ((A X. {(/)}) i^i (C X. {1o})) = (/)
2		xp01disj 5188	. . . 4 |- ((B X. {(/)}) i^i (D X. {1o})) = (/)
3		unen 5493	. . . 4 |- ((((A X. {(/)}) ~~ (B X. {(/)}) /\ (C X. {1o}) ~~ (D X. {1o})) /\ (((A X. {(/)}) i^i (C X. {1o})) = (/) /\ ((B X. {(/)}) i^i (D X. {1o})) = (/))) -> ((A X. {(/)}) u. (C X. {1o})) ~~ ((B X. {(/)}) u. (D X. {1o})))
4	1, 2, 3	mpanr12 778	. . 3 |- (((A X. {(/)}) ~~ (B X. {(/)}) /\ (C X. {1o}) ~~ (D X. {1o})) -> ((A X. {(/)}) u. (C X. {1o})) ~~ ((B X. {(/)}) u. (D X. {1o})))
5		cdaen.1	. . . . 5 |- A e. _V
6		0ex 3446	. . . . . 6 |- (/) e. _V
7	5, 6	xpsnen 5494	. . . . 5 |- (A X. {(/)}) ~~ A
8		enen1 5540	. . . . 5 |- ((A e. _V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)})))
9	5, 7, 8	mp2an 761	. . . 4 |- ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)}))
10		cdaen.2	. . . . 5 |- B e. _V
11	10, 6	xpsnen 5494	. . . . 5 |- (B X. {(/)}) ~~ B
12		enen2 5541	. . . . 5 |- ((B e. _V /\ (B X. {(/)}) ~~ B) -> (A ~~ (B X. {(/)}) <-> A ~~ B))
13	10, 11, 12	mp2an 761	. . . 4 |- (A ~~ (B X. {(/)}) <-> A ~~ B)
14	9, 13	bitri 190	. . 3 |- ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ B)
15		cdaen.3	. . . . 5 |- C e. _V
16		1on 5182	. . . . . . 7 |- 1o e. On
17	16	elisseti 2301	. . . . . 6 |- 1o e. _V
18	15, 17	xpsnen 5494	. . . . 5 |- (C X. {1o}) ~~ C
19		enen1 5540	. . . . 5 |- ((C e. _V /\ (C X. {1o}) ~~ C) -> ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o})))
20	15, 18, 19	mp2an 761	. . . 4 |- ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o}))
21		cdaen.4	. . . . 5 |- D e. _V
22	21, 17	xpsnen 5494	. . . . 5 |- (D X. {1o}) ~~ D
23		enen2 5541	. . . . 5 |- ((D e. _V /\ (D X. {1o}) ~~ D) -> (C ~~ (D X. {1o}) <-> C ~~ D))
24	21, 22, 23	mp2an 761	. . . 4 |- (C ~~ (D X. {1o}) <-> C ~~ D)
25	20, 24	bitri 190	. . 3 |- ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ D)
26	4, 14, 25	syl2anbr 505	. 2 |- ((A ~~ B /\ C ~~ D) -> ((A X. {(/)}) u. (C X. {1o})) ~~ ((B X. {(/)}) u. (D X. {1o})))
27	5, 15	cdavali 6068	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
28	10, 21	cdavali 6068	. 2 |- (B +c D) = ((B X. {(/)}) u. (D X. {1o}))
29	26, 27, 28	3brtr4g 3369	1 |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Oncon0 3657   X. cxp 3984  (class class class)co 4884  1oc1o 5172   ~~ cen 5423   +c ccda 6065

This theorem is referenced by:  cdaeng 6074  nnacda 6088

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-er 5318  df-en 5427  df-cda 6066

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