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Mirrors > Home > MPE Home > Th. List > tskun | Structured version Visualization version GIF version |
Description: The union of two elements of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tskun | ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ∪ 𝐵) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniprg 4685 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
2 | 1 | 3adant1 1121 | . 2 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
3 | simp1l 1211 | . . 3 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
4 | simp1r 1212 | . . 3 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → Tr 𝑇) | |
5 | tskpr 9927 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) | |
6 | 5 | 3adant1r 1180 | . . 3 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) |
7 | tskuni 9940 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ {𝐴, 𝐵} ∈ 𝑇) → ∪ {𝐴, 𝐵} ∈ 𝑇) | |
8 | 3, 4, 6, 7 | syl3anc 1439 | . 2 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → ∪ {𝐴, 𝐵} ∈ 𝑇) |
9 | 2, 8 | eqeltrrd 2859 | 1 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ∪ 𝐵) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∪ cun 3789 {cpr 4399 ∪ cuni 4671 Tr wtr 4987 Tarskictsk 9905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-ac2 9620 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-smo 7726 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-oi 8704 df-har 8752 df-r1 8924 df-card 9098 df-aleph 9099 df-cf 9100 df-acn 9101 df-ac 9272 df-wina 9841 df-ina 9842 df-tsk 9906 |
This theorem is referenced by: (None) |
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