Theorem cdaen 4924

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 4143	. . . . 5 |- ((A X. {(/)}) i^i (C X. {1o})) = (/)
2		xp01disj 4143	. . . . 5 |- ((B X. {(/)}) i^i (D X. {1o})) = (/)
3	1, 2	pm3.2i 285	. . . 4 |- (((A X. {(/)}) i^i (C X. {1o})) = (/) /\ ((B X. {(/)}) i^i (D X. {1o})) = (/))
4		unen 4434	. . . 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})))
5	3, 4	mpan2 696	. . 3 |- (((A X. {(/)}) ~~ (B X. {(/)}) /\ (C X. {1o}) ~~ (D X. {1o})) -> ((A X. {(/)}) u. (C X. {1o})) ~~ ((B X. {(/)}) u. (D X. {1o})))
6		cdaen.1	. . . . 5 |- A e. V
7		0ex 2711	. . . . . 6 |- (/) e. V
8	6, 7	xpsnen 4435	. . . . 5 |- (A X. {(/)}) ~~ A
9		enen1 4477	. . . . 5 |- ((A e. V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)})))
10	6, 8, 9	mp2an 697	. . . 4 |- ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)}))
11		cdaen.2	. . . . 5 |- B e. V
12	11, 7	xpsnen 4435	. . . . 5 |- (B X. {(/)}) ~~ B
13		enen2 4478	. . . . 5 |- ((B e. V /\ (B X. {(/)}) ~~ B) -> (A ~~ (B X. {(/)}) <-> A ~~ B))
14	11, 12, 13	mp2an 697	. . . 4 |- (A ~~ (B X. {(/)}) <-> A ~~ B)
15	10, 14	bitr 173	. . 3 |- ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ B)
16		cdaen.3	. . . . 5 |- C e. V
17		1on 4138	. . . . . . 7 |- 1o e. On
18	17	elisseti 1818	. . . . . 6 |- 1o e. V
19	16, 18	xpsnen 4435	. . . . 5 |- (C X. {1o}) ~~ C
20		enen1 4477	. . . . 5 |- ((C e. V /\ (C X. {1o}) ~~ C) -> ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o})))
21	16, 19, 20	mp2an 697	. . . 4 |- ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o}))
22		cdaen.4	. . . . 5 |- D e. V
23	22, 18	xpsnen 4435	. . . . 5 |- (D X. {1o}) ~~ D
24		enen2 4478	. . . . 5 |- ((D e. V /\ (D X. {1o}) ~~ D) -> (C ~~ (D X. {1o}) <-> C ~~ D))
25	22, 23, 24	mp2an 697	. . . 4 |- (C ~~ (D X. {1o}) <-> C ~~ D)
26	21, 25	bitr 173	. . 3 |- ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ D)
27	5, 15, 26	syl2anbr 456	. 2 |- ((A ~~ B /\ C ~~ D) -> ((A X. {(/)}) u. (C X. {1o})) ~~ ((B X. {(/)}) u. (D X. {1o})))
28	6, 16	cdaval 4920	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
29	11, 22	cdaval 4920	. 2 |- (B +c D) = ((B X. {(/)}) u. (D X. {1o}))
30	27, 28, 29	3brtr4g 2647	1 |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409   class class class wbr 2619  Oncon0 2948   X. cxp 3168  (class class class)co 3963  1oc1o 4128   ~~ cen 4364   +c ccda 4917

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

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-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  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-suc 2954  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-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-1o 4133  df-er 4261  df-en 4368  df-cda 4918

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