MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  entr3i Unicode version

Theorem entr3i 7002
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
Hypotheses
Ref Expression
entr3i.1  |-  A  ~~  B
entr3i.2  |-  A  ~~  C
Assertion
Ref Expression
entr3i  |-  B  ~~  C

Proof of Theorem entr3i
StepHypRef Expression
1 entr3i.1 . . 3  |-  A  ~~  B
21ensymi 6996 . 2  |-  B  ~~  A
3 entr3i.2 . 2  |-  A  ~~  C
42, 3entri 7000 1  |-  B  ~~  C
Colors of variables: wff set class
Syntax hints:   class class class wbr 4102    ~~ cen 6945
This theorem is referenced by:  xpomenOLD  12580  cpnnen  12598
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-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  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-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-er 6744  df-en 6949
  Copyright terms: Public domain W3C validator