MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frirr Structured version   Visualization version   GIF version

Theorem frirr 5654
Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
frirr ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem frirr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → 𝑅 Fr 𝐴)
2 snssi 4812 . . . 4 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
32adantl 483 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
4 snnzg 4779 . . . 4 (𝐵𝐴 → {𝐵} ≠ ∅)
54adantl 483 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ≠ ∅)
6 snex 5432 . . . 4 {𝐵} ∈ V
76frc 5643 . . 3 ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
81, 3, 5, 7syl3anc 1372 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9 breq1 5152 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
109rabeq0w 4384 . . . . . 6 ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦)
11 breq2 5153 . . . . . . . 8 (𝑦 = 𝐵 → (𝑧𝑅𝑦𝑧𝑅𝐵))
1211notbid 318 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐵))
1312ralbidv 3178 . . . . . 6 (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
1410, 13bitrid 283 . . . . 5 (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
1514rexsng 4679 . . . 4 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
16 breq1 5152 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
1716notbid 318 . . . . 5 (𝑧 = 𝐵 → (¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1817ralsng 4678 . . . 4 (𝐵𝐴 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1915, 18bitrd 279 . . 3 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
2019adantl 483 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
218, 20mpbid 231 1 ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  wss 3949  c0 4323  {csn 4629   class class class wbr 5149   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-fr 5632
This theorem is referenced by:  efrirr  5658  predfrirr  6336  dfwe2  7761  bnj1417  34052  efrunt  34682  ifr0  43209
  Copyright terms: Public domain W3C validator