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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snssiALTVD | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of snssiALT 44848. (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 3968 | . . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) | |
| 2 | idn1 44594 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
| 3 | idn2 44633 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ {𝐴} ) | |
| 4 | velsn 4642 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 5 | 3, 4 | e2bi 44652 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 = 𝐴 ) |
| 6 | eleq1a 2836 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
| 7 | 2, 5, 6 | e12 44744 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ {𝐴} ▶ 𝑥 ∈ 𝐵 ) |
| 8 | 7 | in2 44625 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ) |
| 9 | 8 | gen11 44636 | . . 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 44711 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
| 12 | 11 | in1 44591 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-sn 4627 df-vd1 44590 df-vd2 44598 |
| This theorem is referenced by: (None) |
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