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Theorem pmtrfmvdn0 18089

Description: A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)

**Hypotheses**
Ref	Expression
pmtrrn.t	⊢ 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r	⊢ 𝑅 = ran 𝑇

**Assertion**
Ref	Expression
pmtrfmvdn0	⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅)

**Proof of Theorem pmtrfmvdn0**
Step	Hyp	Ref	Expression
1		2on0 7723	. 2 ⊢ 2_𝑜 ≠ ∅
2		pmtrrn.t	. . . . . . . 8 ⊢ 𝑇 = (pmTrsp‘𝐷)
3		pmtrrn.r	. . . . . . . 8 ⊢ 𝑅 = ran 𝑇
4		eqid 2771	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
5	2, 3, 4	pmtrfrn 18085	. . . . . . 7 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2_𝑜) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
6	5	simpld 482	. . . . . 6 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2_𝑜))
7	6	simp3d 1138	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2_𝑜)
8		enen1 8256	. . . . 5 ⊢ (dom (𝐹 ∖ I ) ≈ 2_𝑜 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2_𝑜 ≈ ∅))
9	7, 8	syl 17	. . . 4 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2_𝑜 ≈ ∅))
10		en0 8172	. . . 4 ⊢ (dom (𝐹 ∖ I ) ≈ ∅ ↔ dom (𝐹 ∖ I ) = ∅)
11		en0 8172	. . . 4 ⊢ (2_𝑜 ≈ ∅ ↔ 2_𝑜 = ∅)
12	9, 10, 11	3bitr3g 302	. . 3 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = ∅ ↔ 2_𝑜 = ∅))
13	12	necon3bid 2987	. 2 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≠ ∅ ↔ 2_𝑜 ≠ ∅))
14	1, 13	mpbiri 248	1 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∖ cdif 3720 ⊆ wss 3723 ∅c0 4063 class class class wbr 4786 I cid 5156 dom cdm 5249 ran crn 5250 ‘cfv 6031 2_𝑜c2o 7707 ≈ cen 8106 pmTrspcpmtr 18068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-1o 7713 df-2o 7714 df-er 7896 df-en 8110 df-fin 8113 df-pmtr 18069

This theorem is referenced by: psgnunilem3 18123

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