Theorem cdaen 4904

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 4133	. . . . 5 |- ((A X. {(/)}) i^i (C X. {1o})) = (/)
2		xp01disj 4133	. . . . 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 4420	. . . 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 695	. . 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 2706	. . . . . 6 |- (/) e. V
8	6, 7	xpsnen 4421	. . . . 5 |- (A X. {(/)}) ~~ A
9		enen1 4463	. . . . 5 |- ((A e. V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)})))
10	6, 8, 9	mp2an 696	. . . 4 |- ((A X. {(/)}) ~~ (B X. {(/)}) <-> A ~~ (B X. {(/)}))
11		cdaen.2	. . . . 5 |- B e. V
12	11, 7	xpsnen 4421	. . . . 5 |- (B X. {(/)}) ~~ B
13		enen2 4464	. . . . 5 |- ((B e. V /\ (B X. {(/)}) ~~ B) -> (A ~~ (B X. {(/)}) <-> A ~~ B))
14	11, 12, 13	mp2an 696	. . . 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 4128	. . . . . . 7 |- 1o e. On
18	17	elisseti 1814	. . . . . 6 |- 1o e. V
19	16, 18	xpsnen 4421	. . . . 5 |- (C X. {1o}) ~~ C
20		enen1 4463	. . . . 5 |- ((C e. V /\ (C X. {1o}) ~~ C) -> ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o})))
21	16, 19, 20	mp2an 696	. . . 4 |- ((C X. {1o}) ~~ (D X. {1o}) <-> C ~~ (D X. {1o}))
22		cdaen.4	. . . . 5 |- D e. V
23	22, 18	xpsnen 4421	. . . . 5 |- (D X. {1o}) ~~ D
24		enen2 4464	. . . . 5 |- ((D e. V /\ (D X. {1o}) ~~ D) -> (C ~~ (D X. {1o}) <-> C ~~ D))
25	22, 23, 24	mp2an 696	. . . 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 4900	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
29	11, 22	cdaval 4900	. 2 |- (B +c D) = ((B X. {(/)}) u. (D X. {1o}))
30	27, 28, 29	3brtr4g 2642	1 |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807   u. cun 2041   i^i cin 2042  (/)c0 2276  {csn 2405   class class class wbr 2614  Oncon0 2943   X. cxp 3163  (class class class)co 3954  1oc1o 4118   ~~ cen 4354   +c ccda 4897

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-opr 3956  df-oprab 3957  df-1o 4123  df-er 4251  df-en 4357  df-cda 4898

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