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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwtrALT | Structured version Visualization version GIF version |
Description: Virtual deduction proof of sspwtr 44071. This proof is the same as the proof of sspwtr 44071 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 5257 | . . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
2 | simpr 484 | . . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
3 | ssel 3967 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴)) | |
4 | elpwi 4601 | . . . . . 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 482 | . . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦) | |
8 | 6, 7 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)) |
9 | ssel 3967 | . . . . 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 1923 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
12 | biimpr 219 | . . 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 43727 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∈ wcel 2098 ⊆ wss 3940 𝒫 cpw 4594 Tr wtr 5255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3947 df-ss 3957 df-pw 4596 df-uni 4900 df-tr 5256 |
This theorem is referenced by: (None) |
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