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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 5179. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5179 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 3957 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) | |
2 | idn1 40915 | . . . . . . 7 ⊢ ( Tr 𝐴 ▶ Tr 𝐴 ) | |
3 | idn2 40954 | . . . . . . 7 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
4 | trss 5183 | . . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
5 | 2, 3, 4 | e12 41065 | . . . . . 6 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ⊆ 𝐴 ) |
6 | vex 3499 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | elpw 4545 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
8 | 5, 7 | e2bir 40974 | . . . . 5 ⊢ ( Tr 𝐴 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝒫 𝐴 ) |
9 | 8 | in2 40946 | . . . 4 ⊢ ( Tr 𝐴 ▶ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
10 | 9 | gen11 40957 | . . 3 ⊢ ( Tr 𝐴 ▶ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) ) |
11 | biimpr 222 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴)) | |
12 | 1, 10, 11 | e01 41032 | . 2 ⊢ ( Tr 𝐴 ▶ 𝐴 ⊆ 𝒫 𝐴 ) |
13 | 12 | in1 40912 | 1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 Tr wtr 5174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-in 3945 df-ss 3954 df-pw 4543 df-uni 4841 df-tr 5175 df-vd1 40911 df-vd2 40919 |
This theorem is referenced by: (None) |
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