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Theorem pm54.43lem 7648
Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7617), so that their  A  e.  1 means, in our notation,  A  e.  {
x  |  ( card `  x )  =  1o }. Here we show that this is equivalent to  A  ~~  1o so that we can use the latter more convenient notation in pm54.43 7649. (Contributed by NM, 4-Nov-2013.)
Assertion
Ref Expression
pm54.43lem  |-  ( A 
~~  1o  <->  A  e.  { x  |  ( card `  x
)  =  1o }
)
Distinct variable group:    x, A

Proof of Theorem pm54.43lem
StepHypRef Expression
1 carden2b 7616 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 6653 . . . . 5  |-  1o  e.  om
3 cardnn 7612 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 8 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2344 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2306 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 197 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 6510 . . . . . . . 8  |-  1o  =/=  (/)
9 df-ne 2461 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
108, 9mpbi 199 . . . . . . 7  |-  -.  1o  =  (/)
11 eqeq1 2302 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
1210, 11mtbiri 294 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
13 ndmfv 5568 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1412, 13nsyl2 119 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
15 1on 6502 . . . . . 6  |-  1o  e.  On
16 onenon 7598 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1715, 16ax-mp 8 . . . . 5  |-  1o  e.  dom  card
18 carden2 7636 . . . . 5  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
1914, 17, 18sylancl 643 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
207, 19mpbid 201 . . 3  |-  ( (
card `  A )  =  1o  ->  A  ~~  1o )
215, 20impbii 180 . 2  |-  ( A 
~~  1o  <->  ( card `  A
)  =  1o )
22 elex 2809 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2314, 22syl 15 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
24 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2524eqeq1d 2304 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2623, 25elab3 2934 . 2  |-  ( A  e.  { x  |  ( card `  x
)  =  1o }  <->  (
card `  A )  =  1o )
2721, 26bitr4i 243 1  |-  ( A 
~~  1o  <->  A  e.  { x  |  ( card `  x
)  =  1o }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   Oncon0 4408   omcom 4672   dom cdm 4705   ` cfv 5271   1oc1o 6488    ~~ cen 6876   cardccrd 7584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588
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