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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 7334 | . . . 4 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | 3 | eleq1d 2899 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
5 | 1, 4 | mtbiri 329 | . 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 398 = wceq 1537 ∈ wcel 2114 ∅c0 4293 × cxp 5555 dom cdm 5557 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: ndmovass 7338 ndmovdistr 7339 ndmovord 7340 ndmovordi 7341 caovmo 7387 brecop2 8393 eceqoveq 8404 addcanpi 10323 mulcanpi 10324 ordpipq 10366 recmulnq 10388 recclnq 10390 ltexnq 10399 nsmallnq 10401 ltbtwnnq 10402 prlem934 10457 ltaddpr 10458 ltaddpr2 10459 ltexprlem2 10461 ltexprlem3 10462 ltexprlem4 10463 ltexprlem6 10465 ltexprlem7 10466 addcanpr 10470 prlem936 10471 mappsrpr 10532 |
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