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Mirrors > Home > MPE Home > Th. List > Mathboxes > snssl | Structured version Visualization version GIF version |
Description: If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4259. The proof of this theorem was automatically generated from snsslVD 38086 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snssl.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snssl | ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssl.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | snid 4155 | . 2 ⊢ 𝐴 ∈ {𝐴} |
3 | ssel2 3563 | . 2 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵) | |
4 | 2, 3 | mpan2 703 | 1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-sn 4126 |
This theorem is referenced by: (None) |
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