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Theorem frirr 5051
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → 𝑅 Fr 𝐴)
2 snssi 4308 . . . 4 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
32adantl 482 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
4 snnzg 4278 . . . 4 (𝐵𝐴 → {𝐵} ≠ ∅)
54adantl 482 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ≠ ∅)
6 snex 4869 . . . 4 {𝐵} ∈ V
76frc 5040 . . 3 ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
81, 3, 5, 7syl3anc 1323 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9 rabeq0 3931 . . . . . 6 ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10 breq2 4617 . . . . . . . 8 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 308 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 2980 . . . . . 6 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
139, 12syl5bb 272 . . . . 5 (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
1413rexsng 4190 . . . 4 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15 breq1 4616 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐵𝐵𝑅𝐵))
1615notbid 308 . . . . 5 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1716ralsng 4189 . . . 4 (𝐵𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1814, 17bitrd 268 . . 3 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
1918adantl 482 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
208, 19mpbid 222 1 ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3555  c0 3891  {csn 4148   class class class wbr 4613   Fr wfr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-fr 5033
This theorem is referenced by:  efrirr  5055  predfrirr  5668  dfwe2  6928  bnj1417  30814  efrunt  31295  ifr0  38133
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