Theorem map1 5489

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 4907	. 2 |- (1o ^m A) e. _V
2		0ex 3446	. . 3 |- (/) e. _V
3	2	a1i 8	. 2 |- (x e. (1o ^m A) -> (/) e. _V)
4		map1.1	. . . 4 |- A e. _V
5		p0ex 3495	. . . 4 |- {(/)} e. _V
6	4, 5	xpex 4096	. . 3 |- (A X. {(/)}) e. _V
7	6	a1i 8	. 2 |- (y e. 1o -> (A X. {(/)}) e. _V)
8		ancom 482	. . 3 |- ((y e. 1o /\ x = (A X. {(/)})) <-> (x = (A X. {(/)}) /\ y e. 1o))
9		df1o2 5185	. . . . . . 7 |- 1o = {(/)}
10	9	opreq1i 4892	. . . . . 6 |- (1o ^m A) = ({(/)} ^m A)
11	10	eleq2i 1961	. . . . 5 |- (x e. (1o ^m A) <-> x e. ({(/)} ^m A))
12	5, 4	elmap 5393	. . . . 5 |- (x e. ({(/)} ^m A) <-> x:A-->{(/)})
13	2	fconst2 4823	. . . . 5 |- (x:A-->{(/)} <-> x = (A X. {(/)}))
14	11, 12, 13	3bitrri 195	. . . 4 |- (x = (A X. {(/)}) <-> x e. (1o ^m A))
15		el1o 5191	. . . 4 |- (y e. 1o <-> y = (/))
16	14, 15	anbi12i 540	. . 3 |- ((x = (A X. {(/)}) /\ y e. 1o) <-> (x e. (1o ^m A) /\ y = (/)))
17	8, 16	bitr2i 191	. 2 |- ((x e. (1o ^m A) /\ y = (/)) <-> (y e. 1o /\ x = (A X. {(/)})))
18	1, 3, 7, 17	en2 5461	1 |- (1o ^m A) ~~ 1o

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044   class class class wbr 3338   X. cxp 3984  -->wf 3994  (class class class)co 4884  1oc1o 5172   ^m cmap 5381   ~~ cen 5423

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-map 5383  df-en 5427

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