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Mirrors > Home > MPE Home > Th. List > Mathboxes > snssiALTVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snssiALT 44826. (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 | df-ss 3980 | . . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
2 | idn1 44572 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | idn2 44611 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ {𝐴} ) | |
4 | velsn 4647 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | 3, 4 | e2bi 44630 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 = 𝐴 ) |
6 | eleq1a 2834 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
7 | 2, 5, 6 | e12 44722 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ 𝐵 ) |
8 | 7 | in2 44603 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ) |
9 | 8 | gen11 44614 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ) |
10 | biimpr 220 | . . 3 ⊢ (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) → {𝐴} ⊆ 𝐵)) | |
11 | 1, 9, 10 | e01 44689 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
12 | 11 | in1 44569 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-sn 4632 df-vd1 44568 df-vd2 44576 |
This theorem is referenced by: (None) |
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