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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 5365, ax-un 7755. (Revised by BTernaryTau, 12-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| sdom1 | ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df1o2 8513 | . . . . . . 7 ⊢ 1o = {∅} | |
| 2 | 1 | breq2i 5151 | . . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅}) | 
| 3 | brdomi 8999 | . . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅}) | |
| 4 | f1cdmsn 7302 | . . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥}) | |
| 5 | vex 3484 | . . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V | |
| 6 | 5 | ensn1 9061 | . . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o | 
| 7 | breq1 5146 | . . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o)) | |
| 8 | 6, 7 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o) | 
| 9 | 8 | exlimiv 1930 | . . . . . . . . . 10 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o) | 
| 10 | 4, 9 | syl 17 | . . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 1o) | 
| 11 | 10 | expcom 413 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o)) | 
| 12 | 11 | exlimdv 1933 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o)) | 
| 13 | 3, 12 | syl5 34 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ {∅} → 𝐴 ≈ 1o)) | 
| 14 | 2, 13 | biimtrid 242 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ 1o → 𝐴 ≈ 1o)) | 
| 15 | iman 401 | . . . . 5 ⊢ ((𝐴 ≼ 1o → 𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
| 16 | 14, 15 | sylib 218 | . . . 4 ⊢ (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | 
| 17 | brsdom 9015 | . . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
| 18 | 16, 17 | sylnibr 329 | . . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o) | 
| 19 | 18 | necon4ai 2972 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) | 
| 20 | 1n0 8526 | . . . 4 ⊢ 1o ≠ ∅ | |
| 21 | 1oex 8516 | . . . . 5 ⊢ 1o ∈ V | |
| 22 | 21 | 0sdom 9147 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) | 
| 23 | 20, 22 | mpbir 231 | . . 3 ⊢ ∅ ≺ 1o | 
| 24 | breq1 5146 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o)) | |
| 25 | 23, 24 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o) | 
| 26 | 19, 25 | impbii 209 | 1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ≠ wne 2940 ∅c0 4333 {csn 4626 class class class wbr 5143 –1-1→wf1 6558 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 | 
| This theorem is referenced by: modom 9280 frgpcyg 21592 | 
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