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Mirrors > Home > MPE Home > Th. List > Mathboxes > snssiALT | Structured version Visualization version GIF version |
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4810. This theorem was automatically generated from snssiALTVD 43573 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snssiALT | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 4643 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | eleq1a 2828 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | biimtrid 241 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) |
4 | 3 | alrimiv 1930 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) |
5 | dfss2 3967 | . 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 {csn 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3954 df-ss 3964 df-sn 4628 |
This theorem is referenced by: (None) |
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