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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 6716 | . . . 4 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | 3 | eleq1d 2672 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
5 | 1, 4 | mtbiri 316 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆) |
6 | 5 | con4i 112 | 1 ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 × cxp 5036 dom cdm 5038 (class class class)co 6549 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: ndmovass 6720 ndmovdistr 6721 ndmovord 6722 ndmovordi 6723 caovmo 6769 brecop2 7728 eceqoveq 7740 addcanpi 9600 mulcanpi 9601 ordpipq 9643 recmulnq 9665 recclnq 9667 ltexnq 9676 nsmallnq 9678 ltbtwnnq 9679 prlem934 9734 ltaddpr 9735 ltaddpr2 9736 ltexprlem2 9738 ltexprlem3 9739 ltexprlem4 9740 ltexprlem6 9742 ltexprlem7 9743 addcanpr 9747 prlem936 9748 mappsrpr 9808 |
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