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Mirrors > Home > MPE Home > Th. List > tsk0 | Structured version Visualization version GIF version |
Description: A nonempty Tarski class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.) |
Ref | Expression |
---|---|
tsk0 | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4338 | . . 3 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇) | |
2 | 0ss 4388 | . . . . . 6 ⊢ ∅ ⊆ 𝑥 | |
3 | tskss 10749 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇) | |
4 | 2, 3 | mp3an3 1446 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → ∅ ∈ 𝑇) |
5 | 4 | expcom 413 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
6 | 5 | exlimiv 1925 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
8 | 7 | impcom 407 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ⊆ wss 3940 ∅c0 4314 Tarskictsk 10739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-tsk 10740 |
This theorem is referenced by: tsk1 10755 tskr1om 10758 |
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