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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 5801 | . . . 4 ⊢ Rel ∅ | |
2 | eqid 2728 | . . . . 5 ⊢ ∅ = ∅ | |
3 | fn0 6686 | . . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
4 | 2, 3 | mpbir 230 | . . . 4 ⊢ ∅ Fn ∅ |
5 | tposfn2 8253 | . . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ tpos ∅ Fn ◡∅ |
7 | cnv0 6145 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 7 | fneq2i 6652 | . . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅) |
9 | 6, 8 | mpbi 229 | . 2 ⊢ tpos ∅ Fn ∅ |
10 | fn0 6686 | . 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅) | |
11 | 9, 10 | mpbi 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 8230 |
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 8231 |
This theorem is referenced by: oppchomfval 17693 oppchomfvalOLD 17694 oppgplusfval 19298 opprmulfval 20274 |
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