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Theorem pm54.43lem 7870
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 7839), 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 7871. (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 7838 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 6868 . . . . 5  |-  1o  e.  om
3 cardnn 7834 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 8 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2478 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2440 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 198 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 6725 . . . . . . . 8  |-  1o  =/=  (/)
9 df-ne 2595 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
108, 9mpbi 200 . . . . . . 7  |-  -.  1o  =  (/)
11 eqeq1 2436 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
1210, 11mtbiri 295 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
13 ndmfv 5741 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1412, 13nsyl2 121 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
15 1on 6717 . . . . . 6  |-  1o  e.  On
16 onenon 7820 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1715, 16ax-mp 8 . . . . 5  |-  1o  e.  dom  card
18 carden2 7858 . . . . 5  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
1914, 17, 18sylancl 644 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
207, 19mpbid 202 . . 3  |-  ( (
card `  A )  =  1o  ->  A  ~~  1o )
215, 20impbii 181 . 2  |-  ( A 
~~  1o  <->  ( card `  A
)  =  1o )
22 elex 2951 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2314, 22syl 16 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
24 fveq2 5714 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2524eqeq1d 2438 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2623, 25elab3 3076 . 2  |-  ( A  e.  { x  |  ( card `  x
)  =  1o }  <->  (
card `  A )  =  1o )
2721, 26bitr4i 244 1  |-  ( A 
~~  1o  <->  A  e.  { x  |  ( card `  x
)  =  1o }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2416    =/= wne 2593   _Vcvv 2943   (/)c0 3615   class class class wbr 4199   Oncon0 4568   omcom 4831   dom cdm 4864   ` cfv 5440   1oc1o 6703    ~~ cen 7092   cardccrd 7806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-1o 6710  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-card 7810
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