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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 7321 | . . . 4 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | 3 | eleq1d 2894 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
5 | 1, 4 | mtbiri 328 | . 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 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 × cxp 5546 dom cdm 5548 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: ndmovass 7325 ndmovdistr 7326 ndmovord 7327 ndmovordi 7328 caovmo 7374 brecop2 8380 eceqoveq 8391 addcanpi 10309 mulcanpi 10310 ordpipq 10352 recmulnq 10374 recclnq 10376 ltexnq 10385 nsmallnq 10387 ltbtwnnq 10388 prlem934 10443 ltaddpr 10444 ltaddpr2 10445 ltexprlem2 10447 ltexprlem3 10448 ltexprlem4 10449 ltexprlem6 10451 ltexprlem7 10452 addcanpr 10456 prlem936 10457 mappsrpr 10518 |
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