Theorem cda1en 6076

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 3446	. . . . . 6 |- (/) e. _V
3	1, 2	xpsnen 5494	. . . . 5 |- (A X. {(/)}) ~~ A
4		cardid 5977	. . . . 5 |- (card` A) ~~ A
5	1, 3, 4	entr4i 5478	. . . 4 |- (A X. {(/)}) ~~ (card` A)
6		1on 5182	. . . . . 6 |- 1o e. On
7	6	elisseti 2301	. . . . 5 |- 1o e. _V
8	7, 7	xpsnen 5494	. . . . 5 |- (1o X. {1o}) ~~ 1o
9		fvex 4689	. . . . . 6 |- (card` A) e. _V
10	9	ensn1 5483	. . . . 5 |- {(card` A)} ~~ 1o
11	7, 8, 10	entr4i 5478	. . . 4 |- (1o X. {1o}) ~~ {(card` A)}
12	5, 11	pm3.2i 307	. . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13		xp01disj 5188	. . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
14		cardon 5976	. . . . . 6 |- (card` A) e. On
15	14	onordi 3774	. . . . 5 |- Ord (card` A)
16		orddisj 3701	. . . . 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 307	. . 3 |- (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))
19		unen 5493	. . 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 761	. 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
21	1, 7	cdavali 6068	. 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
22		df-suc 3663	. 2 |- suc (card` A) = ((card` A) u. {(card` A)})
23	20, 21, 22	3brtr4i 3365	1 |- (A +c 1o) ~~ suc (card` A)

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Ord word 3656  Oncon0 3657  suc csuc 3659   X. cxp 3984  ` cfv 3998  (class class class)co 4884  1oc1o 5172   ~~ cen 5423  cardccrd 5859   +c ccda 6065

This theorem is referenced by:  nnacda 6088

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-er 5318  df-en 5427  df-card 5862  df-cda 6066

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