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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.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 12-Dec-2024.) |
Ref | Expression |
---|---|
sdom1 | ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8424 | . . . . . . 7 ⊢ 1o = {∅} | |
2 | 1 | breq2i 5118 | . . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅}) |
3 | brdomi 8905 | . . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅}) | |
4 | f1cdmsn 7233 | . . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥}) | |
5 | vex 3452 | . . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V | |
6 | 5 | ensn1 8968 | . . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o |
7 | breq1 5113 | . . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o)) | |
8 | 6, 7 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o) |
9 | 8 | exlimiv 1934 | . . . . . . . . . 10 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o) |
10 | 4, 9 | syl 17 | . . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 1o) |
11 | 10 | expcom 415 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o)) |
12 | 11 | exlimdv 1937 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o)) |
13 | 3, 12 | syl5 34 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ {∅} → 𝐴 ≈ 1o)) |
14 | 2, 13 | biimtrid 241 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ 1o → 𝐴 ≈ 1o)) |
15 | iman 403 | . . . . 5 ⊢ ((𝐴 ≼ 1o → 𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
16 | 14, 15 | sylib 217 | . . . 4 ⊢ (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) |
17 | brsdom 8922 | . . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
18 | 16, 17 | sylnibr 329 | . . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o) |
19 | 18 | necon4ai 2976 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
20 | 1n0 8439 | . . . 4 ⊢ 1o ≠ ∅ | |
21 | 1oex 8427 | . . . . 5 ⊢ 1o ∈ V | |
22 | 21 | 0sdom 9058 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
23 | 20, 22 | mpbir 230 | . . 3 ⊢ ∅ ≺ 1o |
24 | breq1 5113 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o)) | |
25 | 23, 24 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o) |
26 | 19, 25 | impbii 208 | 1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ≠ wne 2944 ∅c0 4287 {csn 4591 class class class wbr 5110 –1-1→wf1 6498 1oc1o 8410 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: modom 9195 frgpcyg 20996 |
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