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Mirrors > Home > MPE Home > Th. List > tskssel | Structured version Visualization version GIF version |
Description: A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tskssel | ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomnen 8973 | . . 3 ⊢ (𝐴 ≺ 𝑇 → ¬ 𝐴 ≈ 𝑇) | |
2 | 1 | 3ad2ant3 1136 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → ¬ 𝐴 ≈ 𝑇) |
3 | tsken 10745 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)) | |
4 | 3 | 3adant3 1133 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)) |
5 | 4 | ord 863 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → (¬ 𝐴 ≈ 𝑇 → 𝐴 ∈ 𝑇)) |
6 | 2, 5 | mpd 15 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 846 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3947 class class class wbr 5147 ≈ cen 8932 ≺ csdm 8934 Tarskictsk 10739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-sdom 8938 df-tsk 10740 |
This theorem is referenced by: tskpr 10761 tskwe2 10764 tskord 10771 tskcard 10772 tskurn 10780 |
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