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Mirrors > Home > ILE Home > Th. List > tpos0 | GIF version |
Description: Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tpos0 | ⊢ tpos ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4751 | . . . 4 ⊢ Rel ∅ | |
2 | eqid 2177 | . . . . 5 ⊢ ∅ = ∅ | |
3 | fn0 5335 | . . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
4 | 2, 3 | mpbir 146 | . . . 4 ⊢ ∅ Fn ∅ |
5 | tposfn2 6266 | . . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅)) | |
6 | 1, 4, 5 | mp2 16 | . . 3 ⊢ tpos ∅ Fn ◡∅ |
7 | cnv0 5032 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 7 | fneq2i 5311 | . . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅) |
9 | 6, 8 | mpbi 145 | . 2 ⊢ tpos ∅ Fn ∅ |
10 | fn0 5335 | . 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅) | |
11 | 9, 10 | mpbi 145 | 1 ⊢ tpos ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∅c0 3422 ◡ccnv 4625 Rel wrel 4631 Fn wfn 5211 tpos ctpos 6244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-tpos 6245 |
This theorem is referenced by: (None) |
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