New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  intasym GIF version

Theorem intasym 5028
 Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 9-Sep-2004.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
intasym ((RR) I ↔ xy((xRy yRx) → x = y))
Distinct variable group:   x,y,R

Proof of Theorem intasym
StepHypRef Expression
1 ssrel 4844 . 2 ((RR) I ↔ xy(x, y (RR) → x, y I ))
2 df-br 4640 . . . . 5 (x(RR)yx, y (RR))
3 brin 4693 . . . . . 6 (x(RR)y ↔ (xRy xRy))
4 brcnv 4892 . . . . . . 7 (xRyyRx)
54anbi2i 675 . . . . . 6 ((xRy xRy) ↔ (xRy yRx))
63, 5bitri 240 . . . . 5 (x(RR)y ↔ (xRy yRx))
72, 6bitr3i 242 . . . 4 (x, y (RR) ↔ (xRy yRx))
8 df-br 4640 . . . . 5 (x I yx, y I )
9 vex 2862 . . . . . 6 y V
109ideq 4870 . . . . 5 (x I yx = y)
118, 10bitr3i 242 . . . 4 (x, y I ↔ x = y)
127, 11imbi12i 316 . . 3 ((x, y (RR) → x, y I ) ↔ ((xRy yRx) → x = y))
13122albii 1567 . 2 (xy(x, y (RR) → x, y I ) ↔ xy((xRy yRx) → x = y))
141, 13bitri 240 1 ((RR) I ↔ xy((xRy yRx) → x = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257  ⟨cop 4561   class class class wbr 4639   I cid 4763  ◡ccnv 4771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-id 4767  df-cnv 4785 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator