Theorem cda1en 5078

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 2785	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 4576	. . . . 5 |- (A X. {(/)}) ~~ A
4		cardid 4975	. . . . 5 |- (card` A) ~~ A
5	1, 3, 4	entr4i 4560	. . . 4 |- (A X. {(/)}) ~~ (card` A)
6		1on 4274	. . . . . 6 |- 1o e. On
7	6	elisseti 1864	. . . . 5 |- 1o e. V
8	7, 7	xpsnen 4576	. . . . 5 |- (1o X. {1o}) ~~ 1o
9		fvex 3843	. . . . . 6 |- (card` A) e. V
10	9	ensn1 4565	. . . . 5 |- {(card` A)} ~~ 1o
11	7, 8, 10	entr4i 4560	. . . 4 |- (1o X. {1o}) ~~ {(card` A)}
12	5, 11	pm3.2i 283	. . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13		xp01disj 4279	. . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
14		cardon 4974	. . . . . 6 |- (card` A) e. On
15	14	onordi 3074	. . . . 5 |- Ord (card` A)
16		orddisj 3013	. . . . 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 283	. . 3 |- (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))
19		unen 4575	. . 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 701	. 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
21	1, 7	cdavali 5070	. 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
22		df-suc 2981	. 2 |- suc (card` A) = ((card` A) u. {(card` A)})
23	20, 21, 22	3brtr4i 2716	1 |- (A +c 1o) ~~ suc (card` A)

Syntax hints:   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467   class class class wbr 2692  Ord word 2974  Oncon0 2975  suc csuc 2977   X. cxp 3249  ` cfv 3263  (class class class)co 4021  1oc1o 4264   ~~ cen 4505  cardccrd 4959   +c ccda 5067

This theorem is referenced by:  nnacda 5090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1o 4269  df-er 4401  df-en 4509  df-card 4962  df-cda 5068

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