MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-1st Structured version   Unicode version

Theorem ex-1st 25367
Description: Example for df-1st 6773. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
Assertion
Ref Expression
ex-1st  |-  ( 1st `  <. 3 ,  4
>. )  =  3

Proof of Theorem ex-1st
StepHypRef Expression
1 3re 10605 . . 3  |-  3  e.  RR
21elexi 3116 . 2  |-  3  e.  _V
3 4re 10608 . . 3  |-  4  e.  RR
43elexi 3116 . 2  |-  4  e.  _V
52, 4op1st 6781 1  |-  ( 1st `  <. 3 ,  4
>. )  =  3
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   <.cop 4022   ` cfv 5570   1stc1st 6771   RRcr 9480   3c3 10582   4c4 10583
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  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
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-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-1st 6773  df-2 10590  df-3 10591  df-4 10592
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator