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Theorem pm54.43lem 7516
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 7485), 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 7517. (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 7484 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 6523 . . . . 5  |-  1o  e.  om
3 cardnn 7480 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 10 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2301 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2263 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 199 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 6380 . . . . . . . 8  |-  1o  =/=  (/)
9 df-ne 2414 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
108, 9mpbi 201 . . . . . . 7  |-  -.  1o  =  (/)
11 eqeq1 2259 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
1210, 11mtbiri 296 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
13 ndmfv 5405 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1412, 13nsyl2 121 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
15 1on 6372 . . . . . 6  |-  1o  e.  On
16 onenon 7466 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1715, 16ax-mp 10 . . . . 5  |-  1o  e.  dom  card
18 carden2 7504 . . . . 5  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
1914, 17, 18sylancl 646 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
207, 19mpbid 203 . . 3  |-  ( (
card `  A )  =  1o  ->  A  ~~  1o )
215, 20impbii 182 . 2  |-  ( A 
~~  1o  <->  ( card `  A
)  =  1o )
22 elex 2735 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2314, 22syl 17 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
24 fveq2 5377 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2524eqeq1d 2261 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2623, 25elab3 2858 . 2  |-  ( A  e.  { x  |  ( card `  x
)  =  1o }  <->  (
card `  A )  =  1o )
2721, 26bitr4i 245 1  |-  ( A 
~~  1o  <->  A  e.  { x  |  ( card `  x
)  =  1o }
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    = wceq 1619    e. wcel 1621   {cab 2239    =/= wne 2412   _Vcvv 2727   (/)c0 3362   class class class wbr 3920   Oncon0 4285   omcom 4547   dom cdm 4580   ` cfv 4592   1oc1o 6358    ~~ cen 6746   cardccrd 7452
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1o 6365  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-card 7456
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