Theorem en1b 7576

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)

Assertion

Ref	Expression
en1b	|- ( A ~~ 1o <-> A = { U. A } )

Proof of Theorem en1b

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 7575	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
2		id 22	. . . . 5 |- ( A = { x } -> A = { x } )
3		unieq 4243	. . . . . . 7 |- ( A = { x } -> U. A = U. { x } )
4		vex 3109	. . . . . . . 8 |- x e. _V
5	4	unisn 4250	. . . . . . 7 |- U. { x } = x
6	3, 5	syl6eq 2511	. . . . . 6 |- ( A = { x } -> U. A = x )
7	6	sneqd 4028	. . . . 5 |- ( A = { x } -> { U. A } = { x } )
8	2, 7	eqtr4d 2498	. . . 4 |- ( A = { x } -> A = { U. A } )
9	8	exlimiv 1727	. . 3 |- ( E. x A = { x } -> A = { U. A } )
10	1, 9	sylbi 195	. 2 |- ( A ~~ 1o -> A = { U. A } )
11		id 22	. . 3 |- ( A = { U. A } -> A = { U. A } )
12		snex 4678	. . . . . 6 |- { U. A } e. _V
13	11, 12	syl6eqel 2550	. . . . 5 |- ( A = { U. A } -> A e. _V )
14		uniexg 6570	. . . . 5 |- ( A e. _V -> U. A e. _V )
15	13, 14	syl 16	. . . 4 |- ( A = { U. A } -> U. A e. _V )
16		ensn1g 7573	. . . 4 |- ( U. A e. _V -> { U. A } ~~ 1o )
17	15, 16	syl 16	. . 3 |- ( A = { U. A } -> { U. A } ~~ 1o )
18	11, 17	eqbrtrd 4459	. 2 |- ( A = { U. A } -> A ~~ 1o )
19	10, 18	impbii 188	1 |- ( A ~~ 1o <-> A = { U. A } )

Syntax hints:   <-> wb 184   = wceq 1398   E.wex 1617   e. wcel 1823   _Vcvv 3106   {csn 4016   U.cuni 4235   class class class wbr 4439   1oc1o 7115   ~~ cen 7506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-en 7510

This theorem is referenced by:  en1uniel  7580  sylow2alem2  16837  sylow2a  16838  frgpcyg  18785  ptcmplem3  20720  cnextfvval  20731  cnextcn  20733  minveclem4a  22011  isppw  23586  xrge0tsmsbi  28011