Theorem cda1en 3721

Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.

Hypothesis

Ref	Expression
cda0en.1	|- A e. V

Assertion

Ref	Expression
cda1en	|- (A +c 1o) ~~ suc (card` A)

Proof of Theorem cda1en

Step	Hyp	Ref	Expression
1		cda0en.1	. . . . 5 |- A e. V
2		0ex 1745	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 3339	. . . . 5 |- (A X. {(/)}) ~~ A
4		cardid 3635	. . . . 5 |- (card` A) ~~ A
5	1, 3, 4	entr4 3324	. . . 4 |- (A X. {(/)}) ~~ (card` A)
6		1o 3109	. . . . . 6 |- 1o e. On
7	6	elisseti 1355	. . . . 5 |- 1o e. V
8	7, 7	xpsnen 3339	. . . . 5 |- (1o X. {1o}) ~~ 1o
9		fvex 2838	. . . . . 6 |- (card` A) e. V
10	9	ensn1 3329	. . . . 5 |- {(card` A)} ~~ 1o
11	7, 8, 10	entr4 3324	. . . 4 |- (1o X. {1o}) ~~ {(card` A)}
12	5, 11	pm3.2i 234	. . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13		0ne1oOLD 3113	. . . . 5 |- -. (/) = 1o
14		xpsndisj 2655	. . . . 5 |- (-. (/) = 1o -> ((A X. {(/)}) i^i (1o X. {1o})) = (/))
15	13, 14	ax-mp 6	. . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
16		cardon 3634	. . . . . 6 |- (card` A) e. On
17	16	onord 2343	. . . . 5 |- Ord (card` A)
18		orddisj 2236	. . . . 5 |- (Ord (card` A) -> ((card` A) i^i {(card` A)}) = (/))
19	17, 18	ax-mp 6	. . . 4 |- ((card` A) i^i {(card` A)}) = (/)
20	15, 19	pm3.2i 234	. . 3 |- (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))
21		unen 3338	. . 3 |- ((((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)}) /\ (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))) -> ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)}))
22	12, 20, 21	mp2an 520	. 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
23	1, 7	cdaval 3717	. 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
24		df-suc 2205	. 2 |- suc (card` A) = ((card` A) u. {(card` A)})
25	22, 23, 24	3brtr4 2085	1 |- (A +c 1o) ~~ suc (card` A)

Syntax hints:  -. wn 1   /\ 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  Ord word 2198  Oncon0 2199  suc csuc 2201   X. cxp 2408  ` cfv 2422  (class class class)co 3001  1oc1o 3099   ~~ cen 3271  cardccrd 3620   +c ccda 3714

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  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  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-card 3623  df-cda 3715

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