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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trsspwALT | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of the left-to-right implication of dftr4 5266. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5266 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| trsspwALT | ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ss 3968 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) | |
| 2 | idn1 44594 | . . . . . . 7 ⊢ ( Tr 𝐴 ▶ Tr 𝐴 ) | |
| 3 | idn2 44633 | . . . . . . 7 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
| 4 | trss 5270 | . . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
| 5 | 2, 3, 4 | e12 44744 | . . . . . 6 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ⊆ 𝐴 ) |
| 6 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 7 | 6 | elpw 4604 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
| 8 | 5, 7 | e2bir 44653 | . . . . 5 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝒫 𝐴 ) |
| 9 | 8 | in2 44625 | . . . 4 ⊢ ( Tr 𝐴 ▶ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
| 10 | 9 | gen11 44636 | . . 3 ⊢ ( Tr 𝐴 ▶ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
| 11 | biimpr 220 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴)) | |
| 12 | 1, 10, 11 | e01 44711 | . 2 ⊢ ( Tr 𝐴 ▶ 𝐴 ⊆ 𝒫 𝐴 ) |
| 13 | 12 | in1 44591 | 1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀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-ral 3062 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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