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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 4734. This theorem was automatically generated from snssiALTVD 41154 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 4576 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | eleq1a 2908 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | syl5bi 244 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) |
4 | 3 | alrimiv 1924 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) |
5 | dfss2 3954 | . 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-sn 4561 |
This theorem is referenced by: (None) |
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