Theorem cda1en 4926

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 2711	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 4435	. . . . 5 |- (A X. {(/)}) ~~ A
4		cardid 4828	. . . . 5 |- (card` A) ~~ A
5	1, 3, 4	entr4 4419	. . . 4 |- (A X. {(/)}) ~~ (card` A)
6		1on 4138	. . . . . 6 |- 1o e. On
7	6	elisseti 1818	. . . . 5 |- 1o e. V
8	7, 7	xpsnen 4435	. . . . 5 |- (1o X. {1o}) ~~ 1o
9		fvex 3732	. . . . . 6 |- (card` A) e. V
10	9	ensn1 4424	. . . . 5 |- {(card` A)} ~~ 1o
11	7, 8, 10	entr4 4419	. . . 4 |- (1o X. {1o}) ~~ {(card` A)}
12	5, 11	pm3.2i 285	. . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13		xp01disj 4143	. . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
14		cardon 4827	. . . . . 6 |- (card` A) e. On
15	14	onord 3095	. . . . 5 |- Ord (card` A)
16		orddisj 2985	. . . . 5 |- (Ord (card` A) -> ((card` A) i^i {(card` A)}) = (/))
17	15, 16	ax-mp 7	. . . 4 |- ((card` A) i^i {(card` A)}) = (/)
18	13, 17	pm3.2i 285	. . 3 |- (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))
19		unen 4434	. . 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)}))
20	12, 18, 19	mp2an 697	. 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
21	1, 7	cdaval 4920	. 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
22		df-suc 2954	. 2 |- suc (card` A) = ((card` A) u. {(card` A)})
23	20, 21, 22	3brtr4 2643	1 |- (A +c 1o) ~~ suc (card` A)

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409   class class class wbr 2619  Ord word 2947  Oncon0 2948  suc csuc 2950   X. cxp 3168  ` cfv 3182  (class class class)co 3963  1oc1o 4128   ~~ cen 4364  cardccrd 4813   +c ccda 4917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-1o 4133  df-er 4261  df-en 4368  df-card 4816  df-cda 4918

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