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Mirrors > Home > MPE Home > Th. List > Mathboxes > snsslVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snssl 41041. (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 40785 | . . 3 ⊢ ( {𝐴} ⊆ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) | |
2 | snsslVD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | snid 4591 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
4 | ssel2 3959 | . . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵) | |
5 | 1, 3, 4 | e10an 40906 | . 2 ⊢ ( {𝐴} ⊆ 𝐵 ▶ 𝐴 ∈ 𝐵 ) |
6 | 5 | in1 40782 | 1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-sn 4558 df-vd1 40781 |
This theorem is referenced by: (None) |
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