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Mirrors > Home > MPE Home > Th. List > 0sdomg | Structured version Visualization version GIF version |
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) |
Ref | Expression |
---|---|
0sdomg | ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0domg 8632 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | |
2 | brsdom 8520 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
3 | 2 | baib 536 | . . 3 ⊢ (∅ ≼ 𝐴 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)) |
5 | ensymb 8545 | . . . 4 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 ≈ ∅) | |
6 | en0 8560 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
7 | 5, 6 | bitri 276 | . . 3 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) |
8 | 7 | necon3bbii 3060 | . 2 ⊢ (¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅) |
9 | 4, 8 | syl6bb 288 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 ≈ cen 8494 ≼ cdom 8495 ≺ csdm 8496 |
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 ax-un 7450 |
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-ne 3014 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-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: 0sdom 8636 fodomr 8656 pwdom 8657 sdom1 8706 infn0 8768 fodomfib 8786 domwdom 9026 iunfictbso 9528 djulepw 9606 fin45 9802 fodomb 9936 brdom3 9938 gchxpidm 10079 inar1 10185 csdfil 22430 ovoliunnul 24035 carsgclctunlem3 31477 domalom 34567 ovoliunnfl 34815 voliunnfl 34817 volsupnfl 34818 ensucne0OLD 39774 nnfoctb 41186 caragenunicl 42683 |
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