Theorem cda1en 4909

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 2707	. . . . . 6 |- (/) e. V
3	1, 2	xpsnen 4424	. . . . 5 |- (A X. {(/)}) ~~ A
4		cardid 4811	. . . . 5 |- (card` A) ~~ A
5	1, 3, 4	entr4 4409	. . . 4 |- (A X. {(/)}) ~~ (card` A)
6		1on 4131	. . . . . 6 |- 1o e. On
7	6	elisseti 1815	. . . . 5 |- 1o e. V
8	7, 7	xpsnen 4424	. . . . 5 |- (1o X. {1o}) ~~ 1o
9		fvex 3727	. . . . . 6 |- (card` A) e. V
10	9	ensn1 4414	. . . . 5 |- {(card` A)} ~~ 1o
11	7, 8, 10	entr4 4409	. . . 4 |- (1o X. {1o}) ~~ {(card` A)}
12	5, 11	pm3.2i 285	. . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13		xp01disj 4136	. . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
14		cardon 4810	. . . . . 6 |- (card` A) e. On
15	14	onord 3091	. . . . 5 |- Ord (card` A)
16		orddisj 2981	. . . . 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 4423	. . 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 696	. 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
21	1, 7	cdaval 4903	. 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
22		df-suc 2950	. 2 |- suc (card` A) = ((card` A) u. {(card` A)})
23	20, 21, 22	3brtr4 2639	1 |- (A +c 1o) ~~ suc (card` A)

Syntax hints:   /\ 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  Ord word 2943  Oncon0 2944  suc csuc 2946   X. cxp 3164  ` cfv 3178  (class class class)co 3958  1oc1o 4121   ~~ cen 4357  cardccrd 4796   +c ccda 4900

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  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  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-reu 1649  df-rab 1650  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-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  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-card 4799  df-cda 4901

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