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Mirrors > Home > MPE Home > Th. List > Mathboxes > snsslVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snssl 41157. (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 40901 | . . 3 ⊢ ( {𝐴} ⊆ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) | |
2 | snsslVD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | snid 4594 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
4 | ssel2 3961 | . . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵) | |
5 | 1, 3, 4 | e10an 41022 | . 2 ⊢ ( {𝐴} ⊆ 𝐵 ▶ 𝐴 ∈ 𝐵 ) |
6 | 5 | in1 40898 | 1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 ⊆ 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 df-vd1 40897 |
This theorem is referenced by: (None) |
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