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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 8108 | . 2 ⊢ 1o = {∅} | |
2 | tsk0 10177 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) | |
3 | tsksn 10174 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ ∅ ∈ 𝑇) → {∅} ∈ 𝑇) | |
4 | 2, 3 | syldan 593 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → {∅} ∈ 𝑇) |
5 | 1, 4 | eqeltrid 2915 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 ≠ wne 3014 ∅c0 4289 {csn 4559 1oc1o 8087 Tarskictsk 10162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-pow 5257 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-suc 6190 df-1o 8094 df-tsk 10163 |
This theorem is referenced by: tsk2 10179 |
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