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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 4260 | . . 3 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇) | |
2 | 0ss 4304 | . . . . . 6 ⊢ ∅ ⊆ 𝑥 | |
3 | tskss 10169 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇) | |
4 | 2, 3 | mp3an3 1447 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → ∅ ∈ 𝑇) |
5 | 4 | expcom 417 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
6 | 5 | exlimiv 1931 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
7 | 1, 6 | sylbi 220 | . 2 ⊢ (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇)) |
8 | 7 | impcom 411 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 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 ax-sep 5167 |
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-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-tsk 10160 |
This theorem is referenced by: tsk1 10175 tskr1om 10178 |
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