Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tsk1 | Structured version Visualization version GIF version |
Description: One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) |
Ref | Expression |
---|---|
tsk1 | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8192 | . 2 ⊢ 1o = {∅} | |
2 | tsk0 10342 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) | |
3 | tsksn 10339 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ ∅ ∈ 𝑇) → {∅} ∈ 𝑇) | |
4 | 2, 3 | syldan 594 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → {∅} ∈ 𝑇) |
5 | 1, 4 | eqeltrid 2835 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 {csn 4527 1oc1o 8173 Tarskictsk 10327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-pow 5243 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-suc 6197 df-1o 8180 df-tsk 10328 |
This theorem is referenced by: tsk2 10344 |
Copyright terms: Public domain | W3C validator |