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| Description: Virtual deduction proof of the right-to-left implication of dftr4 5266. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 44841 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sspwtr | ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dftr2 5261 | . . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
| 2 | idn1 44594 | . . . . . . . 8 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 ▶ 𝐴 ⊆ 𝒫 𝐴 ) | |
| 3 | idn2 44633 | . . . . . . . . 9 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ) | |
| 4 | simpr 484 | . . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
| 5 | 3, 4 | e2 44651 | . . . . . . . 8 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ 𝑦 ∈ 𝐴 ) | 
| 6 | ssel 3977 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴)) | |
| 7 | 2, 5, 6 | e12 44744 | . . . . . . 7 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ 𝑦 ∈ 𝒫 𝐴 ) | 
| 8 | elpwi 4607 | . . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴) | |
| 9 | 7, 8 | e2 44651 | . . . . . 6 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ 𝑦 ⊆ 𝐴 ) | 
| 10 | simpl 482 | . . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦) | |
| 11 | 3, 10 | e2 44651 | . . . . . 6 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ 𝑧 ∈ 𝑦 ) | 
| 12 | ssel 3977 | . . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴)) | |
| 13 | 9, 11, 12 | e22 44691 | . . . . 5 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ▶ 𝑧 ∈ 𝐴 ) | 
| 14 | 13 | in2 44625 | . . . 4 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) ) | 
| 15 | 14 | gen12 44638 | . . 3 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 ▶ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) ) | 
| 16 | biimpr 220 | . . 3 ⊢ ((Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) → Tr 𝐴)) | |
| 17 | 1, 15, 16 | e01 44711 | . 2 ⊢ ( 𝐴 ⊆ 𝒫 𝐴 ▶ Tr 𝐴 ) | 
| 18 | 17 | in1 44591 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 Tr wtr 5259 | 
| 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-pw 4602 df-uni 4908 df-tr 5260 df-vd1 44590 df-vd2 44598 | 
| This theorem is referenced by: (None) | 
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