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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 5371, ax-un 7754. (Revised by BTernaryTau, 12-Dec-2024.) |
Ref | Expression |
---|---|
sdom1 | ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8512 | . . . . . . 7 ⊢ 1o = {∅} | |
2 | 1 | breq2i 5156 | . . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅}) |
3 | brdomi 8998 | . . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅}) | |
4 | f1cdmsn 7302 | . . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥}) | |
5 | vex 3482 | . . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V | |
6 | 5 | ensn1 9060 | . . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o |
7 | breq1 5151 | . . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o)) | |
8 | 6, 7 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o) |
9 | 8 | exlimiv 1928 | . . . . . . . . . 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 1931 | . . . . . . 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 9014 | . . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o)) | |
18 | 16, 17 | sylnibr 329 | . . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o) |
19 | 18 | necon4ai 2970 | . 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅) |
20 | 1n0 8525 | . . . 4 ⊢ 1o ≠ ∅ | |
21 | 1oex 8515 | . . . . 5 ⊢ 1o ∈ V | |
22 | 21 | 0sdom 9146 | . . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅) |
23 | 20, 22 | mpbir 231 | . . 3 ⊢ ∅ ≺ 1o |
24 | breq1 5151 | . . 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 1537 ∃wex 1776 ≠ wne 2938 ∅c0 4339 {csn 4631 class class class wbr 5148 –1-1→wf1 6560 1oc1o 8498 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1o 8505 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: modom 9278 frgpcyg 21610 |
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