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Theorem pm54.43lem 8270
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 8239), 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 8271. (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 8238 . . . 4  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
2 1onn 7178 . . . . 5  |-  1o  e.  om
3 cardnn 8234 . . . . 5  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
42, 3ax-mp 5 . . . 4  |-  ( card `  1o )  =  1o
51, 4syl6eq 2508 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  1o )
64eqeq2i 2469 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
76biimpri 206 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( card `  A )  =  (
card `  1o )
)
8 1n0 7035 . . . . . . . 8  |-  1o  =/=  (/)
98neii 2648 . . . . . . 7  |-  -.  1o  =  (/)
10 eqeq1 2455 . . . . . . 7  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  (/)  <->  1o  =  (/) ) )
119, 10mtbiri 303 . . . . . 6  |-  ( (
card `  A )  =  1o  ->  -.  ( card `  A )  =  (/) )
12 ndmfv 5813 . . . . . 6  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1311, 12nsyl2 127 . . . . 5  |-  ( (
card `  A )  =  1o  ->  A  e. 
dom  card )
14 1on 7027 . . . . . 6  |-  1o  e.  On
15 onenon 8220 . . . . . 6  |-  ( 1o  e.  On  ->  1o  e.  dom  card )
1614, 15ax-mp 5 . . . . 5  |-  1o  e.  dom  card
17 carden2 8258 . . . . 5  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
1813, 16, 17sylancl 662 . . . 4  |-  ( (
card `  A )  =  1o  ->  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
197, 18mpbid 210 . . 3  |-  ( (
card `  A )  =  1o  ->  A  ~~  1o )
205, 19impbii 188 . 2  |-  ( A 
~~  1o  <->  ( card `  A
)  =  1o )
21 elex 3077 . . . 4  |-  ( A  e.  dom  card  ->  A  e.  _V )
2213, 21syl 16 . . 3  |-  ( (
card `  A )  =  1o  ->  A  e. 
_V )
23 fveq2 5789 . . . 4  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
2423eqeq1d 2453 . . 3  |-  ( x  =  A  ->  (
( card `  x )  =  1o  <->  ( card `  A
)  =  1o ) )
2522, 24elab3 3210 . 2  |-  ( A  e.  { x  |  ( card `  x
)  =  1o }  <->  (
card `  A )  =  1o )
2620, 25bitr4i 252 1  |-  ( A 
~~  1o  <->  A  e.  { x  |  ( card `  x
)  =  1o }
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2436   _Vcvv 3068   (/)c0 3735   class class class wbr 4390   Oncon0 4817   dom cdm 4938   ` cfv 5516   omcom 6576   1oc1o 7013    ~~ cen 7407   cardccrd 8206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-om 6577  df-1o 7020  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210
This theorem is referenced by: (None)
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