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Theorem trsspwALT 38865
Description: Virtual deduction proof of the left-to-right implication of dftr4 4748. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4748 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3584 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 idn1 38610 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 38658 . . . . . . 7 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
4 trss 4752 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4e12 38771 . . . . . 6 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
6 vex 3198 . . . . . . 7 𝑥 ∈ V
76elpw 4155 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7e2bir 38678 . . . . 5 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
98in2 38650 . . . 4 (   Tr 𝐴   ▶   (𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
109gen11 38661 . . 3 (   Tr 𝐴   ▶   𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
11 biimpr 210 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11e01 38736 . 2 (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
1312in1 38607 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479  wcel 1988  wss 3567  𝒫 cpw 4149  Tr wtr 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-uni 4428  df-tr 4744  df-vd1 38606  df-vd2 38614
This theorem is referenced by: (None)
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