Theorem map1 4411

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255.

Hypothesis

Ref	Expression
map1.1	|- A e. V

Assertion

Ref	Expression
map1	|- (1o ^m A) ~~ 1o

Proof of Theorem map1

Step	Hyp	Ref	Expression
1		oprex 3968	. 2 |- (1o ^m A) e. V
2		0ex 2701	. . 3 |- (/) e. V
3	2	a1i 8	. 2 |- (x e. (1o ^m A) -> (/) e. V)
4		map1.1	. . . 4 |- A e. V
5		p0ex 2760	. . . 4 |- {(/)} e. V
6	4, 5	xpex 3250	. . 3 |- (A X. {(/)}) e. V
7	6	a1i 8	. 2 |- (y e. 1o -> (A X. {(/)}) e. V)
8		ancom 435	. . 3 |- ((y e. 1o /\ x = (A X. {(/)})) <-> (x = (A X. {(/)}) /\ y e. 1o))
9		df1o2 4124	. . . . . . 7 |- 1o = {(/)}
10	9	opreq1i 3956	. . . . . 6 |- (1o ^m A) = ({(/)} ^m A)
11	10	eleq2i 1530	. . . . 5 |- (x e. (1o ^m A) <-> x e. ({(/)} ^m A))
12	5, 4	elmap 4318	. . . . 5 |- (x e. ({(/)} ^m A) <-> x:A-->{(/)})
13	2	fconst2 3832	. . . . 5 |- (x:A-->{(/)} <-> x = (A X. {(/)}))
14	11, 12, 13	3bitrr 178	. . . 4 |- (x = (A X. {(/)}) <-> x e. (1o ^m A))
15		el1o 4130	. . . 4 |- (y e. 1o <-> y = (/))
16	14, 15	anbi12i 481	. . 3 |- ((x = (A X. {(/)}) /\ y e. 1o) <-> (x e. (1o ^m A) /\ y = (/)))
17	8, 16	bitr2 174	. 2 |- ((x e. (1o ^m A) /\ y = (/)) <-> (y e. 1o /\ x = (A X. {(/)})))
18	1, 3, 7, 17	en2 4383	1 |- (1o ^m A) ~~ 1o

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  {csn 2399   class class class wbr 2609   X. cxp 3158  -->wf 3168  (class class class)co 3948  1oc1o 4112   ^m cm 4306   ~~ cen 4348

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-map 4308  df-en 4351

Copyright terms: Public domain