Theorem cdadom1 4905

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 2701	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 4415	. . . . 5 |- (A X. {(/)}) ~~ A
4		domen1 4459	. . . . 5 |- ((A e. V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)})))
5	1, 3, 4	mp2an 695	. . . 4 |- ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)}))
6		cdacomen.2	. . . . 5 |- B e. V
7	6, 2	xpsnen 4415	. . . . 5 |- (B X. {(/)}) ~~ B
8		domen2 4460	. . . . 5 |- ((B e. V /\ (B X. {(/)}) ~~ B) -> (A ~<_ (B X. {(/)}) <-> A ~<_ B))
9	6, 7, 8	mp2an 695	. . . 4 |- (A ~<_ (B X. {(/)}) <-> A ~<_ B)
10	5, 9	bitr 173	. . 3 |- ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ B)
11		cdaassen.3	. . . . . 6 |- C e. V
12		snex 2740	. . . . . 6 |- {1o} e. V
13	11, 12	xpex 3250	. . . . 5 |- (C X. {1o}) e. V
14		domrefg 4374	. . . . 5 |- ((C X. {1o}) e. V -> (C X. {1o}) ~<_ (C X. {1o}))
15	13, 14	ax-mp 7	. . . 4 |- (C X. {1o}) ~<_ (C X. {1o})
16		xp01disj 4127	. . . . 5 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
17		p0ex 2760	. . . . . . 7 |- {(/)} e. V
18	6, 17	xpex 3250	. . . . . 6 |- (B X. {(/)}) e. V
19	18, 13, 13	undom 4418	. . . . 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})))
20	16, 19	mpan2 694	. . . 4 |- (((A X. {(/)}) ~<_ (B X. {(/)}) /\ (C X. {1o}) ~<_ (C X. {1o})) -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
21	15, 20	mpan2 694	. . 3 |- ((A X. {(/)}) ~<_ (B X. {(/)}) -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
22	10, 21	sylbir 201	. 2 |- (A ~<_ B -> ((A X. {(/)}) u. (C X. {1o})) ~<_ ((B X. {(/)}) u. (C X. {1o})))
23	1, 11	cdaval 4892	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
24	6, 11	cdaval 4892	. 2 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
25	22, 23, 24	3brtr4g 2637	1 |- (A ~<_ B -> (A +c C) ~<_ (B +c C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609   X. cxp 3158  (class class class)co 3948  1oc1o 4112   ~~ cen 4348   ~<_ cdom 4349   +c ccda 4889

This theorem is referenced by:  cdadom2 4906  infdif 7511

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-er 4245  df-en 4351  df-dom 4352  df-cda 4890

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