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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 8631 | . . . . 5 ⊢ (1o ≼ 𝐴 → ¬ 𝐴 ≺ 1o) | |
2 | 1 | con2i 141 | . . . 4 ⊢ (𝐴 ≺ 1o → ¬ 1o ≼ 𝐴) |
3 | 0sdom1dom 8704 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | |
4 | 2, 3 | sylnibr 330 | . . 3 ⊢ (𝐴 ≺ 1o → ¬ ∅ ≺ 𝐴) |
5 | relsdom 8504 | . . . . 5 ⊢ Rel ≺ | |
6 | 5 | brrelex1i 5601 | . . . 4 ⊢ (𝐴 ≺ 1o → 𝐴 ∈ V) |
7 | 0sdomg 8634 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
8 | 7 | necon2bbid 3056 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐴 ≺ 1o → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
10 | 4, 9 | mpbird 258 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
11 | 1n0 8108 | . . . 4 ⊢ 1o ≠ ∅ | |
12 | 1oex 8099 | . . . . 5 ⊢ 1o ∈ V | |
13 | 12 | 0sdom 8636 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
14 | 11, 13 | mpbir 232 | . . 3 ⊢ ∅ ≺ 1o |
15 | breq1 5060 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o)) | |
16 | 14, 15 | mpbiri 259 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o) |
17 | 10, 16 | impbii 210 | 1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 class class class wbr 5057 1oc1o 8084 ≼ 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-3or 1080 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-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: modom 8707 frgpcyg 20648 |
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