MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wdomen2 Structured version   Unicode version

Theorem wdomen2 8021
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
wdomen2  |-  ( A 
~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )

Proof of Theorem wdomen2
StepHypRef Expression
1 id 22 . . 3  |-  ( C  ~<_*  A  ->  C  ~<_*  A )
2 endom 7561 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
3 domwdom 8018 . . . 4  |-  ( A  ~<_  B  ->  A  ~<_*  B )
42, 3syl 16 . . 3  |-  ( A 
~~  B  ->  A  ~<_*  B )
5 wdomtr 8019 . . 3  |-  ( ( C  ~<_*  A  /\  A  ~<_*  B
)  ->  C  ~<_*  B )
61, 4, 5syl2anr 478 . 2  |-  ( ( A  ~~  B  /\  C  ~<_*  A )  ->  C  ~<_*  B )
7 id 22 . . 3  |-  ( C  ~<_*  B  ->  C  ~<_*  B )
8 ensym 7583 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
9 endom 7561 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
10 domwdom 8018 . . . 4  |-  ( B  ~<_  A  ->  B  ~<_*  A )
118, 9, 103syl 20 . . 3  |-  ( A 
~~  B  ->  B  ~<_*  A )
12 wdomtr 8019 . . 3  |-  ( ( C  ~<_*  B  /\  B  ~<_*  A
)  ->  C  ~<_*  A )
137, 11, 12syl2anr 478 . 2  |-  ( ( A  ~~  B  /\  C  ~<_*  B )  ->  C  ~<_*  A )
146, 13impbida 832 1  |-  ( A 
~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4456    ~~ cen 7532    ~<_ cdom 7533    ~<_* cwdom 8001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-wdom 8003
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator