Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5166 | . . . 4 ⊢ Rel ∅ | |
2 | eqid 2610 | . . . . 5 ⊢ ∅ = ∅ | |
3 | fn0 5924 | . . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
4 | 2, 3 | mpbir 220 | . . . 4 ⊢ ∅ Fn ∅ |
5 | tposfn2 7261 | . . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ tpos ∅ Fn ◡∅ |
7 | cnv0 5454 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 7 | fneq2i 5900 | . . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅) |
9 | 6, 8 | mpbi 219 | . 2 ⊢ tpos ∅ Fn ∅ |
10 | fn0 5924 | . 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅) | |
11 | 9, 10 | mpbi 219 | 1 ⊢ tpos ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 ◡ccnv 5037 Rel wrel 5043 Fn wfn 5799 tpos ctpos 7238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-tpos 7239 |
This theorem is referenced by: oppchomfval 16197 oppgplusfval 17601 opprmulfval 18448 |
Copyright terms: Public domain | W3C validator |