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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 8438 | . . . . 5 ⊢ (1o ≼ 𝐴 → ¬ 𝐴 ≺ 1o) | |
2 | 1 | con2i 137 | . . . 4 ⊢ (𝐴 ≺ 1o → ¬ 1o ≼ 𝐴) |
3 | 0sdom1dom 8510 | . . . 4 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | |
4 | 2, 3 | sylnibr 321 | . . 3 ⊢ (𝐴 ≺ 1o → ¬ ∅ ≺ 𝐴) |
5 | relsdom 8312 | . . . . 5 ⊢ Rel ≺ | |
6 | 5 | brrelex1i 5455 | . . . 4 ⊢ (𝐴 ≺ 1o → 𝐴 ∈ V) |
7 | 0sdomg 8441 | . . . . 5 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
8 | 7 | necon2bbid 3005 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐴 ≺ 1o → (𝐴 = ∅ ↔ ¬ ∅ ≺ 𝐴)) |
10 | 4, 9 | mpbird 249 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
11 | 1n0 7920 | . . . 4 ⊢ 1o ≠ ∅ | |
12 | 1oex 7912 | . . . . 5 ⊢ 1o ∈ V | |
13 | 12 | 0sdom 8443 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
14 | 11, 13 | mpbir 223 | . . 3 ⊢ ∅ ≺ 1o |
15 | breq1 4929 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o)) | |
16 | 14, 15 | mpbiri 250 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o) |
17 | 10, 16 | impbii 201 | 1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 Vcvv 3410 ∅c0 4173 class class class wbr 4926 1oc1o 7897 ≼ cdom 8303 ≺ csdm 8304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-om 7396 df-1o 7904 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 |
This theorem is referenced by: modom 8513 frgpcyg 20438 |
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