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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 10189 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) | |
2 | df-2o 8106 | . . 3 ⊢ 2o = suc 1o | |
3 | 1on 8112 | . . . 4 ⊢ 1o ∈ On | |
4 | tsksuc 10187 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ On ∧ 1o ∈ 𝑇) → suc 1o ∈ 𝑇) | |
5 | 3, 4 | mp3an2 1445 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ 𝑇) → suc 1o ∈ 𝑇) |
6 | 2, 5 | eqeltrid 2920 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 1o ∈ 𝑇) → 2o ∈ 𝑇) |
7 | 1, 6 | syldan 593 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 Oncon0 6194 suc csuc 6196 1oc1o 8098 2oc2o 8099 Tarskictsk 10173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 df-on 6198 df-suc 6200 df-1o 8105 df-2o 8106 df-tsk 10174 |
This theorem is referenced by: 2domtsk 10191 |
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