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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 6818 | . . . 4 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | 3 | eleq1d 2686 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
5 | 1, 4 | mtbiri 317 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆) |
6 | 5 | con4i 113 | 1 ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 × cxp 5112 dom cdm 5114 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: ndmovass 6822 ndmovdistr 6823 ndmovord 6824 ndmovordi 6825 caovmo 6871 brecop2 7841 eceqoveq 7853 addcanpi 9721 mulcanpi 9722 ordpipq 9764 recmulnq 9786 recclnq 9788 ltexnq 9797 nsmallnq 9799 ltbtwnnq 9800 prlem934 9855 ltaddpr 9856 ltaddpr2 9857 ltexprlem2 9859 ltexprlem3 9860 ltexprlem4 9861 ltexprlem6 9863 ltexprlem7 9864 addcanpr 9868 prlem936 9869 mappsrpr 9929 |
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