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Mirrors > Home > ILE Home > Th. List > snssb | GIF version |
Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.) |
Ref | Expression |
---|---|
snssb | ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3146 | . 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
2 | velsn 3611 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | 2 | imbi1i 238 | . . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | albii 1470 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
5 | eleq1 2240 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | 5 | pm5.74i 180 | . . . 4 ⊢ ((𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝐴 ∈ 𝐵)) |
7 | 6 | albii 1470 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵)) |
8 | 19.23v 1883 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵)) | |
9 | isset 2745 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 9 | bicomi 132 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ V) |
11 | 10 | imbi1i 238 | . . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵)) |
12 | 7, 8, 11 | 3bitri 206 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵)) |
13 | 1, 4, 12 | 3bitri 206 | 1 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2739 ⊆ wss 3131 {csn 3594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2741 df-in 3137 df-ss 3144 df-sn 3600 |
This theorem is referenced by: snssg 3728 |
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