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Mirrors > Home > MPE Home > Th. List > tsktrss | Structured version Visualization version GIF version |
Description: A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tsktrss | ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1134 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → Tr 𝐴) | |
2 | dftr4 5141 | . . 3 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝒫 𝐴) |
4 | tskpwss 10163 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇) | |
5 | 4 | 3adant2 1128 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇) |
6 | 3, 5 | sstrd 3925 | 1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 Tr wtr 5136 Tarskictsk 10159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-tr 5137 df-tsk 10160 |
This theorem is referenced by: (None) |
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