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Theorem tpos0 7618
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 5445 . . . 4 Rel ∅
2 eqid 2797 . . . . 5 ∅ = ∅
3 fn0 6220 . . . . 5 (∅ Fn ∅ ↔ ∅ = ∅)
42, 3mpbir 223 . . . 4 ∅ Fn ∅
5 tposfn2 7610 . . . 4 (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ∅))
61, 4, 5mp2 9 . . 3 tpos ∅ Fn
7 cnv0 5751 . . . 4 ∅ = ∅
87fneq2i 6195 . . 3 (tpos ∅ Fn ∅ ↔ tpos ∅ Fn ∅)
96, 8mpbi 222 . 2 tpos ∅ Fn ∅
10 fn0 6220 . 2 (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅)
119, 10mpbi 222 1 tpos ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  c0 4113  ccnv 5309  Rel wrel 5315   Fn wfn 6094  tpos ctpos 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107  df-tpos 7588
This theorem is referenced by:  oppchomfval  16685  oppgplusfval  18087  opprmulfval  18938
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