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Theorem tpos0 7898
Description: Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tpos0 tpos ∅ = ∅

Proof of Theorem tpos0
StepHypRef Expression
1 rel0 5646 . . . 4 Rel ∅
2 eqid 2820 . . . . 5 ∅ = ∅
3 fn0 6453 . . . . 5 (∅ Fn ∅ ↔ ∅ = ∅)
42, 3mpbir 233 . . . 4 ∅ Fn ∅
5 tposfn2 7890 . . . 4 (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ∅))
61, 4, 5mp2 9 . . 3 tpos ∅ Fn
7 cnv0 5973 . . . 4 ∅ = ∅
87fneq2i 6425 . . 3 (tpos ∅ Fn ∅ ↔ tpos ∅ Fn ∅)
96, 8mpbi 232 . 2 tpos ∅ Fn ∅
10 fn0 6453 . 2 (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅)
119, 10mpbi 232 1 tpos ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  c0 4267  ccnv 5528  Rel wrel 5534   Fn wfn 6324  tpos ctpos 7867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-fv 6337  df-tpos 7868
This theorem is referenced by:  oppchomfval  16960  oppgplusfval  18452  opprmulfval  19351
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