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Theorem trclfvg 13706
Description: The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.)
Assertion
Ref Expression
trclfvg (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)

Proof of Theorem trclfvg
StepHypRef Expression
1 exmid 431 . 2 (𝑅 ∈ V ∨ ¬ 𝑅 ∈ V)
2 trclfvlb 13699 . . 3 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 fvprc 6152 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
42, 3orim12i 538 . 2 ((𝑅 ∈ V ∨ ¬ 𝑅 ∈ V) → (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅))
51, 4ax-mp 5 1 (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560  c0 3897  cfv 5857  t+ctcl 13674
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-trcl 13676
This theorem is referenced by: (None)
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