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Mirrors > Home > MPE Home > Th. List > tpos0 | Structured version Visualization version GIF version |
Description: Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tpos0 | ⊢ tpos ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 5636 | . . . 4 ⊢ Rel ∅ | |
2 | eqid 2798 | . . . . 5 ⊢ ∅ = ∅ | |
3 | fn0 6451 | . . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
4 | 2, 3 | mpbir 234 | . . . 4 ⊢ ∅ Fn ∅ |
5 | tposfn2 7897 | . . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ tpos ∅ Fn ◡∅ |
7 | cnv0 5966 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 7 | fneq2i 6421 | . . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅) |
9 | 6, 8 | mpbi 233 | . 2 ⊢ tpos ∅ Fn ∅ |
10 | fn0 6451 | . 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅) | |
11 | 9, 10 | mpbi 233 | 1 ⊢ tpos ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 ◡ccnv 5518 Rel wrel 5524 Fn wfn 6319 tpos ctpos 7874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-tpos 7875 |
This theorem is referenced by: oppchomfval 16976 oppgplusfval 18468 opprmulfval 19371 |
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