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Theorem pm54.43lem 7600
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 7569), 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 7601. (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 7568 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 6605 . . . . 5  |-  1o  e.  om
3 cardnn 7564 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 10 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2306 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2268 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 199 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 6462 . . . . . . . 8  |-  1o  =/=  (/)
9 df-ne 2423 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
108, 9mpbi 201 . . . . . . 7  |-  -.  1o  =  (/)
11 eqeq1 2264 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
1210, 11mtbiri 296 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
13 ndmfv 5486 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1412, 13nsyl2 121 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
15 1on 6454 . . . . . 6  |-  1o  e.  On
16 onenon 7550 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1715, 16ax-mp 10 . . . . 5  |-  1o  e.  dom  card
18 carden2 7588 . . . . 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 2771 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2314, 22syl 17 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
24 fveq2 5458 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2524eqeq1d 2266 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2623, 25elab3 2896 . 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 2244    =/= wne 2421   _Vcvv 2763   (/)c0 3430   class class class wbr 3997   Oncon0 4364   omcom 4628   dom cdm 4661   ` cfv 4673   1oc1o 6440    ~~ cen 6828   cardccrd 7536
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1o 6447  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-card 7540
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