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Mirrors > Home > MPE Home > Th. List > Mathboxes > snssiALTVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snssiALT 42073. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snssiALTVD | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3877 | . . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
2 | idn1 41819 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | idn2 41858 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ {𝐴} ) | |
4 | velsn 4547 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | 3, 4 | e2bi 41877 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 = 𝐴 ) |
6 | eleq1a 2829 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
7 | 2, 5, 6 | e12 41969 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ 𝐵 ) |
8 | 7 | in2 41850 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ) |
9 | 8 | gen11 41861 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ) |
10 | biimpr 223 | . . 3 ⊢ (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) → {𝐴} ⊆ 𝐵)) | |
11 | 1, 9, 10 | e01 41936 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
12 | 11 | in1 41816 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 {csn 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 df-in 3864 df-ss 3874 df-sn 4532 df-vd1 41815 df-vd2 41823 |
This theorem is referenced by: (None) |
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