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Theorem pm54.43lem 7627
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 7596), 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 7628. (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 7595 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 6632 . . . . 5  |-  1o  e.  om
3 cardnn 7591 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 10 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2332 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2294 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 199 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 6489 . . . . . . . 8  |-  1o  =/=  (/)
9 df-ne 2449 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
108, 9mpbi 201 . . . . . . 7  |-  -.  1o  =  (/)
11 eqeq1 2290 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
1210, 11mtbiri 296 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
13 ndmfv 5513 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1412, 13nsyl2 121 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
15 1on 6481 . . . . . 6  |-  1o  e.  On
16 onenon 7577 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1715, 16ax-mp 10 . . . . 5  |-  1o  e.  dom  card
18 carden2 7615 . . . . 5  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
1914, 17, 18sylancl 645 . . . 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 2797 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2314, 22syl 17 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
24 fveq2 5485 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2524eqeq1d 2292 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2623, 25elab3 2922 . 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 1624    e. wcel 1685   {cab 2270    =/= wne 2447   _Vcvv 2789   (/)c0 3456   class class class wbr 4024   Oncon0 4391   omcom 4655   dom cdm 4688   ` cfv 5221   1oc1o 6467    ~~ cen 6855   cardccrd 7563
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-1o 6474  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-card 7567
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