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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwtrALT | Structured version Visualization version GIF version |
Description: Virtual deduction proof of sspwtr 41032. This proof is the same as the proof of sspwtr 41032 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspwtrALT | ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 5165 | . . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
2 | simpr 485 | . . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
3 | ssel 3958 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴)) | |
4 | elpwi 4547 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴) | |
5 | 2, 3, 4 | syl56 36 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)) |
6 | idd 24 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))) | |
7 | simpl 483 | . . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦) | |
8 | 6, 7 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)) |
9 | ssel 3958 | . . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴)) | |
10 | 5, 8, 9 | syl6c 70 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
11 | 10 | alrimivv 1920 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
12 | biimpr 221 | . . 3 ⊢ ((Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) → Tr 𝐴)) | |
13 | 1, 11, 12 | mpsyl 68 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
14 | 13 | idiALT 40688 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 Tr wtr 5163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 df-uni 4831 df-tr 5164 |
This theorem is referenced by: (None) |
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