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Theorem cda1en 8039
Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cda1en  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  ( A  +c  1o )  ~~  suc  A )

Proof of Theorem cda1en
StepHypRef Expression
1 enrefg 7125 . . . 4  |-  ( A  e.  V  ->  A  ~~  A )
21adantr 452 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  A  ~~  A )
3 ensn1g 7158 . . . . 5  |-  ( A  e.  V  ->  { A }  ~~  1o )
43ensymd 7144 . . . 4  |-  ( A  e.  V  ->  1o  ~~ 
{ A } )
54adantr 452 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  1o  ~~  { A } )
6 simpr 448 . . . 4  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  -.  A  e.  A )
7 disjsn 3855 . . . 4  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
86, 7sylibr 204 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  ( A  i^i  { A } )  =  (/) )
9 cdaenun 8038 . . 3  |-  ( ( A  ~~  A  /\  1o  ~~  { A }  /\  ( A  i^i  { A } )  =  (/) )  ->  ( A  +c  1o )  ~~  ( A  u.  { A }
) )
102, 5, 8, 9syl3anc 1184 . 2  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  ( A  +c  1o )  ~~  ( A  u.  { A } ) )
11 df-suc 4574 . 2  |-  suc  A  =  ( A  u.  { A } )
1210, 11syl6breqr 4239 1  |-  ( ( A  e.  V  /\  -.  A  e.  A
)  ->  ( A  +c  1o )  ~~  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3305    i^i cin 3306   (/)c0 3615   {csn 3801   class class class wbr 4199   suc csuc 4570  (class class class)co 6067   1oc1o 6703    ~~ cen 7092    +c ccda 8031
This theorem is referenced by:  pm110.643ALT  8042  pwsdompw  8068
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-suc 4574  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-ov 6070  df-oprab 6071  df-mpt2 6072  df-1o 6710  df-er 6891  df-en 7096  df-cda 8032
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