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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 5665 | . . . 4 ⊢ Rel ∅ | |
2 | eqid 2818 | . . . . 5 ⊢ ∅ = ∅ | |
3 | fn0 6472 | . . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
4 | 2, 3 | mpbir 232 | . . . 4 ⊢ ∅ Fn ∅ |
5 | tposfn2 7903 | . . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ tpos ∅ Fn ◡∅ |
7 | cnv0 5992 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 7 | fneq2i 6444 | . . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅) |
9 | 6, 8 | mpbi 231 | . 2 ⊢ tpos ∅ Fn ∅ |
10 | fn0 6472 | . 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅) | |
11 | 9, 10 | mpbi 231 | 1 ⊢ tpos ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∅c0 4288 ◡ccnv 5547 Rel wrel 5553 Fn wfn 6343 tpos ctpos 7880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-tpos 7881 |
This theorem is referenced by: oppchomfval 16972 oppgplusfval 18414 opprmulfval 19304 |
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