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Mirrors > Home > MPE Home > Th. List > sdomdif | Structured version Visualization version GIF version |
Description: The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.) |
Ref | Expression |
---|---|
sdomdif | ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8893 | . . . . . 6 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5689 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
3 | ssdif0 4324 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
4 | ssdomg 8943 | . . . . . . 7 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
5 | domnsym 9046 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
6 | 4, 5 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → ¬ 𝐴 ≺ 𝐵)) |
7 | 3, 6 | biimtrrid 242 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 134 | . . 3 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅)) |
10 | 9 | pm2.43i 52 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ (𝐵 ∖ 𝐴) = ∅) |
11 | 10 | neqned 2947 | 1 ⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 class class class wbr 5106 ≼ cdom 8884 ≺ csdm 8885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 |
This theorem is referenced by: domtriomlem 10383 konigthlem 10509 odcau 19391 |
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