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Mirrors > Home > MPE Home > Th. List > Mathboxes > trsspwALT2 | Structured version Visualization version GIF version |
Description: Virtual deduction proof of trsspwALT 42438. This proof is the same as the proof of trsspwALT 42438 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
trsspwALT2 | ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3907 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) | |
2 | id 22 | . . . . . . 7 ⊢ (Tr 𝐴 → Tr 𝐴) | |
3 | idd 24 | . . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)) | |
4 | trss 5200 | . . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
5 | 2, 3, 4 | sylsyld 61 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) |
6 | vex 3436 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | elpw 4537 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
8 | 5, 7 | syl6ibr 251 | . . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) |
9 | 8 | idiALT 42097 | . . . 4 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) |
10 | 9 | alrimiv 1930 | . . 3 ⊢ (Tr 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) |
11 | biimpr 219 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴)) | |
12 | 1, 10, 11 | mpsyl 68 | . 2 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
13 | 12 | idiALT 42097 | 1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∈ wcel 2106 ⊆ wss 3887 𝒫 cpw 4533 Tr wtr 5191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-v 3434 df-in 3894 df-ss 3904 df-pw 4535 df-uni 4840 df-tr 5192 |
This theorem is referenced by: (None) |
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