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Theorem rankxpu 7635
Description: An upper bound on the rank of a cross product. (Contributed by NM, 18-Sep-2006.)
Hypotheses
Ref Expression
rankxpl.1  |-  A  e. 
_V
rankxpl.2  |-  B  e. 
_V
Assertion
Ref Expression
rankxpu  |-  ( rank `  ( A  X.  B
) )  C_  suc  suc  ( rank `  ( A  u.  B )
)

Proof of Theorem rankxpu
StepHypRef Expression
1 xpsspw 4876 . . 3  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
2 rankxpl.1 . . . . . . 7  |-  A  e. 
_V
3 rankxpl.2 . . . . . . 7  |-  B  e. 
_V
42, 3unex 4597 . . . . . 6  |-  ( A  u.  B )  e. 
_V
54pwex 4272 . . . . 5  |-  ~P ( A  u.  B )  e.  _V
65pwex 4272 . . . 4  |-  ~P ~P ( A  u.  B
)  e.  _V
76rankss 7608 . . 3  |-  ( ( A  X.  B ) 
C_  ~P ~P ( A  u.  B )  -> 
( rank `  ( A  X.  B ) )  C_  ( rank `  ~P ~P ( A  u.  B )
) )
81, 7ax-mp 8 . 2  |-  ( rank `  ( A  X.  B
) )  C_  ( rank `  ~P ~P ( A  u.  B )
)
95rankpw 7602 . . 3  |-  ( rank `  ~P ~P ( A  u.  B ) )  =  suc  ( rank `  ~P ( A  u.  B ) )
104rankpw 7602 . . . 4  |-  ( rank `  ~P ( A  u.  B ) )  =  suc  ( rank `  ( A  u.  B )
)
11 suceq 4536 . . . 4  |-  ( (
rank `  ~P ( A  u.  B )
)  =  suc  ( rank `  ( A  u.  B ) )  ->  suc  ( rank `  ~P ( A  u.  B
) )  =  suc  suc  ( rank `  ( A  u.  B )
) )
1210, 11ax-mp 8 . . 3  |-  suc  ( rank `  ~P ( A  u.  B ) )  =  suc  suc  ( rank `  ( A  u.  B ) )
139, 12eqtri 2378 . 2  |-  ( rank `  ~P ~P ( A  u.  B ) )  =  suc  suc  ( rank `  ( A  u.  B ) )
148, 13sseqtri 3286 1  |-  ( rank `  ( A  X.  B
) )  C_  suc  suc  ( rank `  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226    C_ wss 3228   ~Pcpw 3701   suc csuc 4473    X. cxp 4766   ` cfv 5334   rankcrnk 7522
This theorem is referenced by:  rankxplim3  7638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-reg 7393  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-recs 6472  df-rdg 6507  df-r1 7523  df-rank 7524
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