Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pmtrfmvdn0 | Structured version Visualization version GIF version |
Description: A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
pmtrrn.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtrrn.r | ⊢ 𝑅 = ran 𝑇 |
Ref | Expression |
---|---|
pmtrfmvdn0 | ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 7456 | . 2 ⊢ 2𝑜 ≠ ∅ | |
2 | pmtrrn.t | . . . . . . . 8 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
3 | pmtrrn.r | . . . . . . . 8 ⊢ 𝑅 = ran 𝑇 | |
4 | eqid 2610 | . . . . . . . 8 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
5 | 2, 3, 4 | pmtrfrn 17701 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
6 | 5 | simpld 474 | . . . . . 6 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜)) |
7 | 6 | simp3d 1068 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2𝑜) |
8 | enen1 7985 | . . . . 5 ⊢ (dom (𝐹 ∖ I ) ≈ 2𝑜 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2𝑜 ≈ ∅)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2𝑜 ≈ ∅)) |
10 | en0 7905 | . . . 4 ⊢ (dom (𝐹 ∖ I ) ≈ ∅ ↔ dom (𝐹 ∖ I ) = ∅) | |
11 | en0 7905 | . . . 4 ⊢ (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅) | |
12 | 9, 10, 11 | 3bitr3g 301 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = ∅ ↔ 2𝑜 = ∅)) |
13 | 12 | necon3bid 2826 | . 2 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≠ ∅ ↔ 2𝑜 ≠ ∅)) |
14 | 1, 13 | mpbiri 247 | 1 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 I cid 4948 dom cdm 5038 ran crn 5039 ‘cfv 5804 2𝑜c2o 7441 ≈ cen 7838 pmTrspcpmtr 17684 |
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-rep 4699 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-3or 1032 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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-fin 7845 df-pmtr 17685 |
This theorem is referenced by: psgnunilem3 17739 |
Copyright terms: Public domain | W3C validator |