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Theorem tpos0 8262
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 5801 . . . 4 Rel ∅
2 eqid 2728 . . . . 5 ∅ = ∅
3 fn0 6686 . . . . 5 (∅ Fn ∅ ↔ ∅ = ∅)
42, 3mpbir 230 . . . 4 ∅ Fn ∅
5 tposfn2 8254 . . . 4 (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ∅))
61, 4, 5mp2 9 . . 3 tpos ∅ Fn
7 cnv0 6145 . . . 4 ∅ = ∅
87fneq2i 6652 . . 3 (tpos ∅ Fn ∅ ↔ tpos ∅ Fn ∅)
96, 8mpbi 229 . 2 tpos ∅ Fn ∅
10 fn0 6686 . 2 (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅)
119, 10mpbi 229 1 tpos ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  c0 4323  ccnv 5677  Rel wrel 5683   Fn wfn 6543  tpos ctpos 8231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-tpos 8232
This theorem is referenced by:  oppchomfval  17694  oppchomfvalOLD  17695  oppgplusfval  19299  opprmulfval  20275
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