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| Mirrors > Home > MPE Home > Th. List > tskmcl | Structured version Visualization version GIF version | ||
| Description: A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
| Ref | Expression |
|---|---|
| tskmcl | ⊢ (tarskiMap‘𝐴) ∈ Tarski |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tskmval 10879 | . . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | |
| 2 | ssrab2 4080 | . . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski | |
| 3 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
| 4 | grothtsk 10875 | . . . . . . 7 ⊢ ∪ Tarski = V | |
| 5 | 3, 4 | eleqtrrdi 2852 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski) |
| 6 | eluni2 4911 | . . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
| 7 | 5, 6 | sylib 218 | . . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) |
| 8 | rabn0 4389 | . . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
| 9 | 7, 8 | sylibr 234 | . . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) |
| 10 | inttsk 10814 | . . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski) | |
| 11 | 2, 9, 10 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski) |
| 12 | 1, 11 | eqeltrd 2841 | . 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski) |
| 13 | fvprc 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅) | |
| 14 | 0tsk 10795 | . . 3 ⊢ ∅ ∈ Tarski | |
| 15 | 13, 14 | eqeltrdi 2849 | . 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski) |
| 16 | 12, 15 | pm2.61i 182 | 1 ⊢ (tarskiMap‘𝐴) ∈ Tarski |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ∩ cint 4946 ‘cfv 6561 Tarskictsk 10788 tarskiMapctskm 10877 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-groth 10863 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-tsk 10789 df-tskm 10878 |
| This theorem is referenced by: eltskm 10883 |
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