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Mirrors > Home > MPE Home > Th. List > sdom1 | Structured version Visualization version GIF version |
Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
sdom1 | ⊢ (𝐴 ≺ 1𝑜 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 7971 | . . . . 5 ⊢ (1𝑜 ≼ 𝐴 → ¬ 𝐴 ≺ 1𝑜) | |
2 | 1 | con2i 133 | . . . 4 ⊢ (𝐴 ≺ 1𝑜 → ¬ 1𝑜 ≼ 𝐴) |
3 | 0sdom1dom 8043 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ 1𝑜 ≼ 𝐴) | |
4 | 2, 3 | sylnibr 318 | . . 3 ⊢ (𝐴 ≺ 1𝑜 → ¬ ∅ ≺ 𝐴) |
5 | relsdom 7848 | . . . . 5 ⊢ Rel ≺ | |
6 | 5 | brrelexi 5082 | . . . 4 ⊢ (𝐴 ≺ 1𝑜 → 𝐴 ∈ V) |
7 | 0sdomg 7974 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
8 | 7 | necon2bbid 2825 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐴 ≺ 1𝑜 → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
10 | 4, 9 | mpbird 246 | . 2 ⊢ (𝐴 ≺ 1𝑜 → 𝐴 = ∅) |
11 | 1n0 7462 | . . . 4 ⊢ 1𝑜 ≠ ∅ | |
12 | 1on 7454 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
13 | 12 | elexi 3186 | . . . . 5 ⊢ 1𝑜 ∈ V |
14 | 13 | 0sdom 7976 | . . . 4 ⊢ (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅) |
15 | 11, 14 | mpbir 220 | . . 3 ⊢ ∅ ≺ 1𝑜 |
16 | breq1 4586 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1𝑜 ↔ ∅ ≺ 1𝑜)) | |
17 | 15, 16 | mpbiri 247 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1𝑜) |
18 | 10, 17 | impbii 198 | 1 ⊢ (𝐴 ≺ 1𝑜 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 class class class wbr 4583 Oncon0 5640 1𝑜c1o 7440 ≼ cdom 7839 ≺ csdm 7840 |
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 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: modom 8046 frgpcyg 19741 |
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