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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snmoore | Structured version Visualization version GIF version |
Description: A singleton is a Moore collection. See bj-snmooreb 34408 for a biconditional version. (Contributed by BJ, 10-Apr-2024.) |
Ref | Expression |
---|---|
bj-snmoore | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisng 4859 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
2 | snidg 4601 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
3 | 1, 2 | eqeltrd 2915 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} ∈ {𝐴}) |
4 | df-ne 3019 | . . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅) | |
5 | sssn 4761 | . . . . . 6 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
6 | biorf 933 | . . . . . . 7 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))) | |
7 | 6 | biimpar 480 | . . . . . 6 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴}) |
8 | 4, 5, 7 | syl2anb 599 | . . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴}) |
9 | inteq 4881 | . . . . . . 7 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴}) | |
10 | intsng 4913 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) | |
11 | eqtr 2843 | . . . . . . . 8 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴) | |
12 | 11 | ex 415 | . . . . . . 7 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴)) |
13 | 9, 10, 12 | syl2im 40 | . . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑉 → ∩ 𝑥 = 𝐴)) |
14 | intex 5242 | . . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V) | |
15 | elsng 4583 | . . . . . . . 8 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴)) | |
16 | 14, 15 | sylbi 219 | . . . . . . 7 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴)) |
17 | 16 | biimprd 250 | . . . . . 6 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴})) |
18 | 13, 17 | sylan9r 511 | . . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴})) |
19 | 8, 18 | syldan 593 | . . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴})) |
20 | 19 | ancoms 461 | . . 3 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴})) |
21 | 20 | impcom 410 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ {𝐴}) |
22 | 3, 21 | bj-ismooredr2 34404 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∪ cuni 4840 ∩ cint 4878 Moorecmoore 34397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-sn 4570 df-pr 4572 df-uni 4841 df-int 4879 df-bj-moore 34398 |
This theorem is referenced by: bj-snmooreb 34408 bj-prmoore 34409 |
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