Theorem cdadom1 4916

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 2707	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 4424	. . . . 5 |- (A X. {(/)}) ~~ A
4		domen1 4468	. . . . 5 |- ((A e. V /\ (A X. {(/)}) ~~ A) -> ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)})))
5	1, 3, 4	mp2an 696	. . . 4 |- ((A X. {(/)}) ~<_ (B X. {(/)}) <-> A ~<_ (B X. {(/)}))
6		cdacomen.2	. . . . 5 |- B e. V
7	6, 2	xpsnen 4424	. . . . 5 |- (B X. {(/)}) ~~ B
8		domen2 4469	. . . . 5 |- ((B e. V /\ (B X. {(/)}) ~~ B) -> (A ~<_ (B X. {(/)}) <-> A ~<_ B))
9	6, 7, 8	mp2an 696	. . . 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 2746	. . . . . 6 |- {1o} e. V
13	11, 12	xpex 3256	. . . . 5 |- (C X. {1o}) e. V
14		domrefg 4383	. . . . 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 4136	. . . . 5 |- ((B X. {(/)}) i^i (C X. {1o})) = (/)
17		p0ex 2766	. . . . . . 7 |- {(/)} e. V
18	6, 17	xpex 3256	. . . . . 6 |- (B X. {(/)}) e. V
19	18, 13, 13	undom 4427	. . . . 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 695	. . . 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 695	. . 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 4903	. 2 |- (A +c C) = ((A X. {(/)}) u. (C X. {1o}))
24	6, 11	cdaval 4903	. 2 |- (B +c C) = ((B X. {(/)}) u. (C X. {1o}))
25	22, 23, 24	3brtr4g 2643	1 |- (A ~<_ B -> (A +c C) ~<_ (B +c C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808   u. cun 2042   i^i cin 2043  (/)c0 2277  {csn 2406   class class class wbr 2615   X. cxp 3164  (class class class)co 3958  1oc1o 4121   ~~ cen 4357   ~<_ cdom 4358   +c ccda 4900

This theorem is referenced by:  cdadom2 4917  infdif 7528

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-int 2530  df-br 2616  df-opab 2663  df-id 2831  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-opr 3960  df-oprab 3961  df-1o 4126  df-er 4254  df-en 4360  df-dom 4361  df-cda 4901

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