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Theorem entri 7562
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
Hypotheses
Ref Expression
entri.1  |-  A  ~~  B
entri.2  |-  B  ~~  C
Assertion
Ref Expression
entri  |-  A  ~~  C

Proof of Theorem entri
StepHypRef Expression
1 entri.1 . 2  |-  A  ~~  B
2 entri.2 . 2  |-  B  ~~  C
3 entr 7560 . 2  |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
41, 2, 3mp2an 670 1  |-  A  ~~  C
Colors of variables: wff setvar class
Syntax hints:   class class class wbr 4439    ~~ cen 7506
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-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  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-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-er 7303  df-en 7510
This theorem is referenced by:  entr2i  7563  entr3i  7564  entr4i  7565  infxpenc2  8390  infxpenc2OLD  8394  cfpwsdom  8950  hashxplem  12475  xpnnen  14026  xpomenOLD  14028  qnnen  14031  rpnnen  14044  rexpen  14045  odhash  16793  cygctb  17093  met2ndci  21191  re2ndc  21472  iscmet3  21898  dyadmbl  22175  opnmblALT  22178  mbfimaopnlem  22228  aannenlem3  22892  mblfinlem1  30291  heiborlem3  30549  heibor  30557  irrapx1  31003
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