Theorem cdadom1 3727

Description: Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.

Hypotheses

Ref	Expression
cdacomen.1	|- A e. V
cdacomen.2	|- B e. V
cdaassen.3	|- C e. V

Assertion

Ref	Expression
cdadom1	|- (A ~<_ B -> (A +c C) ~<_ (B +c C))

Proof of Theorem cdadom1

Step	Hyp	Ref	Expression
1		cdacomen.1	. . . . 5 |- A e. V
2		0ex 1745	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 3339	. . . . 5 |- (A X. {(/)}) ~~ A
4		domen1 3377	. . . . 5 |- ((A e. V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)})))
5	1, 3, 4	mp2an 520	. . . 4 |- ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)}))
6		cdacomen.2	. . . . 5 |- B e. V
7	6, 2	xpsnen 3339	. . . . 5 |- (B X. {(/)}) ~~ B
8		domen2 3378	. . . . 5 |- ((B e. V /\ (B X. {(/)}) ~~ B) -> (A ~<_ (B X. {(/)}) <-> A ~<_ B))
9	6, 7, 8	mp2an 520	. . . 4 |- (A ~<_ (B X. {(/)}) <-> A ~<_ B)
10	5, 9	bitr 151	. . 3 |- ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ B)
11		cdaassen.3	. . . . . 6 |- C e. V
12		snex 1859	. . . . . 6 |- {1o} e. V
13	11, 12	xpex 2488	. . . . 5 |- (C X. {1o}) e. V
14		domrefg 3297	. . . . 5 |- ((C X. {1o}) e. V -> (C X. {1o}) ~<_ (C X. {1o}))
15	13, 14	ax-mp 6	. . . 4 |- (C X. {1o}) ~<_ (C X. {1o})
16		0ne1oOLD 3113	. . . . . 6 |- -. (/) = 1o
17		xpsndisj 2655	. . . . . 6 |- (-. (/) = 1o -> ((B X. {(/)}) i^i (C X. {1o})) = (/))
18	16, 17	ax-mp 6	. . . . 5 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
19		p0ex 1885	. . . . . . 7 |- {(/)} e. V
20	6, 19	xpex 2488	. . . . . 6 |- (B X. {(/)}) e. V
21	20, 13, 13	undom 3342	. . . . 5 |- ((((A X. {(/)}) ~<_ (B X. {(/)}) /\ (C X. {1o}) ~<_ (C X. {1o})) /\ ((B X. {(/)}) i^i (C X. {1o})) = (/)) -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
22	18, 21	mpan2 519	. . . 4 |- (((A X. {(/)}) ~<_ (B X. {(/)}) /\ (C X. {1o}) ~<_ (C X. {1o})) -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
23	15, 22	mpan2 519	. . 3 |- ((A X. {(/)}) ~<_ (B X. {(/)}) -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
24	10, 23	sylbir 176	. 2 |- (A ~<_ B -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
25	1, 11	cdaval 3717	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
26	6, 11	cdaval 3717	. 2 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
27	24, 25, 26	3brtr4g 2088	1 |- (A ~<_ B -> (A +c C) ~<_ (B +c C))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   i^i cin 1486  (/)c0 1707  {csn 1808   class class class wbr 2054   X. cxp 2408  (class class class)co 3001  1oc1o 3099   ~~ cen 3271   ~<_ cdom 3272   +c ccda 3714

This theorem is referenced by:  cdadom2 3728  infdif 4948

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-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-suc 2205  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-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-er 3200  df-en 3274  df-dom 3275  df-cda 3715

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