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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 5354, ax-un 7719. (Revised by BTernaryTau, 12-Dec-2024.) |
Ref | Expression |
---|---|
sdom1 | ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8469 | . . . . . . 7 ⊢ 1o = {∅} | |
2 | 1 | breq2i 5147 | . . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅}) |
3 | brdomi 8951 | . . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅}) | |
4 | f1cdmsn 7273 | . . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥}) | |
5 | vex 3470 | . . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V | |
6 | 5 | ensn1 9014 | . . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o |
7 | breq1 5142 | . . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o)) | |
8 | 6, 7 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o) |
9 | 8 | exlimiv 1925 | . . . . . . . . . 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 1928 | . . . . . . 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 401 | . . . . 5 ⊢ ((𝐴 ≼ 1o → 𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
16 | 14, 15 | sylib 217 | . . . 4 ⊢ (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) |
17 | brsdom 8968 | . . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
18 | 16, 17 | sylnibr 329 | . . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o) |
19 | 18 | necon4ai 2964 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
20 | 1n0 8484 | . . . 4 ⊢ 1o ≠ ∅ | |
21 | 1oex 8472 | . . . . 5 ⊢ 1o ∈ V | |
22 | 21 | 0sdom 9104 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
23 | 20, 22 | mpbir 230 | . . 3 ⊢ ∅ ≺ 1o |
24 | breq1 5142 | . . 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 395 = wceq 1533 ∃wex 1773 ≠ wne 2932 ∅c0 4315 {csn 4621 class class class wbr 5139 –1-1→wf1 6531 1oc1o 8455 ≈ cen 8933 ≼ cdom 8934 ≺ csdm 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-1o 8462 df-en 8937 df-dom 8938 df-sdom 8939 |
This theorem is referenced by: modom 9241 frgpcyg 21457 |
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