Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  poirr2 Structured version   GIF version

Theorem poirr2 4660
 Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
Assertion
Ref Expression
poirr2 (𝑅 Po A → (𝑅 ∩ ( I ↾ A)) = ∅)

Proof of Theorem poirr2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4582 . . . 4 Rel ( I ↾ A)
2 relin2 4399 . . . 4 (Rel ( I ↾ A) → Rel (𝑅 ∩ ( I ↾ A)))
31, 2mp1i 10 . . 3 (𝑅 Po A → Rel (𝑅 ∩ ( I ↾ A)))
4 df-br 3756 . . . . 5 (x(𝑅 ∩ ( I ↾ A))y ↔ ⟨x, y (𝑅 ∩ ( I ↾ A)))
5 brin 3802 . . . . 5 (x(𝑅 ∩ ( I ↾ A))y ↔ (x𝑅y x( I ↾ A)y))
64, 5bitr3i 175 . . . 4 (⟨x, y (𝑅 ∩ ( I ↾ A)) ↔ (x𝑅y x( I ↾ A)y))
7 vex 2554 . . . . . . . . 9 y V
87brres 4561 . . . . . . . 8 (x( I ↾ A)y ↔ (x I y x A))
9 poirr 4035 . . . . . . . . . . 11 ((𝑅 Po A x A) → ¬ x𝑅x)
107ideq 4431 . . . . . . . . . . . . 13 (x I yx = y)
11 breq2 3759 . . . . . . . . . . . . 13 (x = y → (x𝑅xx𝑅y))
1210, 11sylbi 114 . . . . . . . . . . . 12 (x I y → (x𝑅xx𝑅y))
1312notbid 591 . . . . . . . . . . 11 (x I y → (¬ x𝑅x ↔ ¬ x𝑅y))
149, 13syl5ibcom 144 . . . . . . . . . 10 ((𝑅 Po A x A) → (x I y → ¬ x𝑅y))
1514expimpd 345 . . . . . . . . 9 (𝑅 Po A → ((x A x I y) → ¬ x𝑅y))
1615ancomsd 256 . . . . . . . 8 (𝑅 Po A → ((x I y x A) → ¬ x𝑅y))
178, 16syl5bi 141 . . . . . . 7 (𝑅 Po A → (x( I ↾ A)y → ¬ x𝑅y))
1817con2d 554 . . . . . 6 (𝑅 Po A → (x𝑅y → ¬ x( I ↾ A)y))
19 imnan 623 . . . . . 6 ((x𝑅y → ¬ x( I ↾ A)y) ↔ ¬ (x𝑅y x( I ↾ A)y))
2018, 19sylib 127 . . . . 5 (𝑅 Po A → ¬ (x𝑅y x( I ↾ A)y))
2120pm2.21d 549 . . . 4 (𝑅 Po A → ((x𝑅y x( I ↾ A)y) → ⟨x, y ∅))
226, 21syl5bi 141 . . 3 (𝑅 Po A → (⟨x, y (𝑅 ∩ ( I ↾ A)) → ⟨x, y ∅))
233, 22relssdv 4375 . 2 (𝑅 Po A → (𝑅 ∩ ( I ↾ A)) ⊆ ∅)
24 ss0 3251 . 2 ((𝑅 ∩ ( I ↾ A)) ⊆ ∅ → (𝑅 ∩ ( I ↾ A)) = ∅)
2523, 24syl 14 1 (𝑅 Po A → (𝑅 ∩ ( I ↾ A)) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  ⟨cop 3370   class class class wbr 3755   I cid 4016   Po wpo 4022   ↾ cres 4290  Rel wrel 4293 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-xp 4294  df-rel 4295  df-res 4300 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator