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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snsslVD | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of snssl 44855. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| snsslVD.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| snsslVD | ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idn1 44599 | . . 3 ⊢ ( {𝐴} ⊆ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) | |
| 2 | snsslVD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 3 | 2 | snid 4661 | . . 3 ⊢ 𝐴 ∈ {𝐴} | 
| 4 | ssel2 3977 | . . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵) | |
| 5 | 1, 3, 4 | e10an 44720 | . 2 ⊢ ( {𝐴} ⊆ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | 
| 6 | 5 | in1 44596 | 1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-sn 4626 df-vd1 44595 | 
| This theorem is referenced by: (None) | 
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