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| Mirrors > Home > MPE Home > Th. List > ndmovrcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.) | 
| Ref | Expression | 
|---|---|
| ndmov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) | 
| ndmovrcl.3 | ⊢ ¬ ∅ ∈ 𝑆 | 
| Ref | Expression | 
|---|---|
| ndmovrcl | ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ndmovrcl.3 | . . 3 ⊢ ¬ ∅ ∈ 𝑆 | |
| 2 | ndmov.1 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
| 3 | 2 | ndmov 7618 | . . . 4 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) | 
| 4 | 3 | eleq1d 2825 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) | 
| 5 | 1, 4 | mtbiri 327 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆) | 
| 6 | 5 | con4i 114 | 1 ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∅c0 4332 × cxp 5682 dom cdm 5684 (class class class)co 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: ndmovass 7622 ndmovdistr 7623 ndmovord 7624 ndmovordi 7625 caovmo 7671 brecop2 8852 eceqoveq 8863 addcanpi 10940 mulcanpi 10941 ordpipq 10983 recmulnq 11005 recclnq 11007 ltexnq 11016 nsmallnq 11018 ltbtwnnq 11019 prlem934 11074 ltaddpr 11075 ltaddpr2 11076 ltexprlem2 11078 ltexprlem3 11079 ltexprlem4 11080 ltexprlem6 11082 ltexprlem7 11083 addcanpr 11087 prlem936 11088 mappsrpr 11149 | 
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