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Mirrors > Home > MPE Home > Th. List > tsk2 | Structured version Visualization version GIF version |
Description: Two is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tsk2 | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsk1 10781 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) | |
2 | df-2o 8481 | . . 3 ⊢ 2o = suc 1o | |
3 | 1on 8492 | . . . 4 ⊢ 1o ∈ On | |
4 | tsksuc 10779 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ On ∧ 1o ∈ 𝑇) → suc 1o ∈ 𝑇) | |
5 | 3, 4 | mp3an2 1446 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ 𝑇) → suc 1o ∈ 𝑇) |
6 | 2, 5 | eqeltrid 2833 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ 𝑇) → 2o ∈ 𝑇) |
7 | 1, 6 | syldan 590 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ≠ wne 2936 ∅c0 4318 Oncon0 6363 suc csuc 6365 1oc1o 8473 2oc2o 8474 Tarskictsk 10765 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 df-suc 6369 df-1o 8480 df-2o 8481 df-tsk 10766 |
This theorem is referenced by: 2domtsk 10783 |
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