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Theorem hfun 30063
Description: The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
hfun  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e. Hf  )

Proof of Theorem hfun
StepHypRef Expression
1 rankung 30051 . . 3  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) )
2 elhf2g 30061 . . . . 5  |-  ( A  e. Hf  ->  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )
)
32ibi 241 . . . 4  |-  ( A  e. Hf  ->  ( rank `  A )  e.  om )
4 elhf2g 30061 . . . . 5  |-  ( B  e. Hf  ->  ( B  e. Hf 
<->  ( rank `  B
)  e.  om )
)
54ibi 241 . . . 4  |-  ( B  e. Hf  ->  ( rank `  B )  e.  om )
6 eleq1a 2537 . . . . . 6  |-  ( (
rank `  B )  e.  om  ->  ( (
( rank `  A )  u.  ( rank `  B
) )  =  (
rank `  B )  ->  ( ( rank `  A
)  u.  ( rank `  B ) )  e. 
om ) )
76adantl 464 . . . . 5  |-  ( ( ( rank `  A
)  e.  om  /\  ( rank `  B )  e.  om )  ->  (
( ( rank `  A
)  u.  ( rank `  B ) )  =  ( rank `  B
)  ->  ( ( rank `  A )  u.  ( rank `  B
) )  e.  om ) )
8 uncom 3634 . . . . . . . . . 10  |-  ( (
rank `  B )  u.  ( rank `  A
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
98eqeq1i 2461 . . . . . . . . 9  |-  ( ( ( rank `  B
)  u.  ( rank `  A ) )  =  ( rank `  A
)  <->  ( ( rank `  A )  u.  ( rank `  B ) )  =  ( rank `  A
) )
109biimpi 194 . . . . . . . 8  |-  ( ( ( rank `  B
)  u.  ( rank `  A ) )  =  ( rank `  A
)  ->  ( ( rank `  A )  u.  ( rank `  B
) )  =  (
rank `  A )
)
1110eleq1d 2523 . . . . . . 7  |-  ( ( ( rank `  B
)  u.  ( rank `  A ) )  =  ( rank `  A
)  ->  ( (
( rank `  A )  u.  ( rank `  B
) )  e.  om  <->  (
rank `  A )  e.  om ) )
1211biimprcd 225 . . . . . 6  |-  ( (
rank `  A )  e.  om  ->  ( (
( rank `  B )  u.  ( rank `  A
) )  =  (
rank `  A )  ->  ( ( rank `  A
)  u.  ( rank `  B ) )  e. 
om ) )
1312adantr 463 . . . . 5  |-  ( ( ( rank `  A
)  e.  om  /\  ( rank `  B )  e.  om )  ->  (
( ( rank `  B
)  u.  ( rank `  A ) )  =  ( rank `  A
)  ->  ( ( rank `  A )  u.  ( rank `  B
) )  e.  om ) )
14 nnord 6681 . . . . . . 7  |-  ( (
rank `  A )  e.  om  ->  Ord  ( rank `  A ) )
15 nnord 6681 . . . . . . 7  |-  ( (
rank `  B )  e.  om  ->  Ord  ( rank `  B ) )
16 ordtri2or2 4963 . . . . . . 7  |-  ( ( Ord  ( rank `  A
)  /\  Ord  ( rank `  B ) )  -> 
( ( rank `  A
)  C_  ( rank `  B )  \/  ( rank `  B )  C_  ( rank `  A )
) )
1714, 15, 16syl2an 475 . . . . . 6  |-  ( ( ( rank `  A
)  e.  om  /\  ( rank `  B )  e.  om )  ->  (
( rank `  A )  C_  ( rank `  B
)  \/  ( rank `  B )  C_  ( rank `  A ) ) )
18 ssequn1 3660 . . . . . . 7  |-  ( (
rank `  A )  C_  ( rank `  B
)  <->  ( ( rank `  A )  u.  ( rank `  B ) )  =  ( rank `  B
) )
19 ssequn1 3660 . . . . . . 7  |-  ( (
rank `  B )  C_  ( rank `  A
)  <->  ( ( rank `  B )  u.  ( rank `  A ) )  =  ( rank `  A
) )
2018, 19orbi12i 519 . . . . . 6  |-  ( ( ( rank `  A
)  C_  ( rank `  B )  \/  ( rank `  B )  C_  ( rank `  A )
)  <->  ( ( (
rank `  A )  u.  ( rank `  B
) )  =  (
rank `  B )  \/  ( ( rank `  B
)  u.  ( rank `  A ) )  =  ( rank `  A
) ) )
2117, 20sylib 196 . . . . 5  |-  ( ( ( rank `  A
)  e.  om  /\  ( rank `  B )  e.  om )  ->  (
( ( rank `  A
)  u.  ( rank `  B ) )  =  ( rank `  B
)  \/  ( (
rank `  B )  u.  ( rank `  A
) )  =  (
rank `  A )
) )
227, 13, 21mpjaod 379 . . . 4  |-  ( ( ( rank `  A
)  e.  om  /\  ( rank `  B )  e.  om )  ->  (
( rank `  A )  u.  ( rank `  B
) )  e.  om )
233, 5, 22syl2an 475 . . 3  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( ( rank `  A )  u.  ( rank `  B
) )  e.  om )
241, 23eqeltrd 2542 . 2  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( rank `  ( A  u.  B
) )  e.  om )
25 unexg 6574 . . 3  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e.  _V )
26 elhf2g 30061 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  (
( A  u.  B
)  e. Hf  <->  ( rank `  ( A  u.  B )
)  e.  om )
)
2725, 26syl 16 . 2  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( ( A  u.  B )  e. Hf 
<->  ( rank `  ( A  u.  B )
)  e.  om )
)
2824, 27mpbird 232 1  |-  ( ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e. Hf  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459    C_ wss 3461   Ord word 4866   ` cfv 5570   omcom 6673   rankcrnk 8172   Hf chf 30057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-r1 8173  df-rank 8174  df-hf 30058
This theorem is referenced by:  hfadj  30065
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