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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 4685. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4685 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 | dfss2 3557 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) | |
2 | idn1 37811 | . . . . . . 7 ⊢ ( Tr 𝐴 ▶ Tr 𝐴 ) | |
3 | idn2 37859 | . . . . . . 7 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
4 | trss 4689 | . . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
5 | 2, 3, 4 | e12 37972 | . . . . . 6 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ⊆ 𝐴 ) |
6 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | elpw 4114 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
8 | 5, 7 | e2bir 37879 | . . . . 5 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝒫 𝐴 ) |
9 | 8 | in2 37851 | . . . 4 ⊢ ( Tr 𝐴 ▶ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
10 | 9 | gen11 37862 | . . 3 ⊢ ( Tr 𝐴 ▶ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
11 | biimpr 209 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴)) | |
12 | 1, 10, 11 | e01 37937 | . 2 ⊢ ( Tr 𝐴 ▶ 𝐴 ⊆ 𝒫 𝐴 ) |
13 | 12 | in1 37808 | 1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 Tr wtr 4680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-tr 4681 df-vd1 37807 df-vd2 37815 |
This theorem is referenced by: (None) |
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