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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 10255 | . . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | |
2 | ssrab2 4055 | . . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski | |
3 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
4 | grothtsk 10251 | . . . . . . 7 ⊢ ∪ Tarski = V | |
5 | 3, 4 | eleqtrrdi 2924 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski) |
6 | eluni2 4835 | . . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
7 | 5, 6 | sylib 220 | . . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) |
8 | rabn0 4338 | . . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
9 | 7, 8 | sylibr 236 | . . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) |
10 | inttsk 10190 | . . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski) | |
11 | 2, 9, 10 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski) |
12 | 1, 11 | eqeltrd 2913 | . 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski) |
13 | fvprc 6657 | . . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅) | |
14 | 0tsk 10171 | . . 3 ⊢ ∅ ∈ Tarski | |
15 | 13, 14 | eqeltrdi 2921 | . 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski) |
16 | 12, 15 | pm2.61i 184 | 1 ⊢ (tarskiMap‘𝐴) ∈ Tarski |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 ∩ cint 4868 ‘cfv 6349 Tarskictsk 10164 tarskiMapctskm 10253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-groth 10239 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-tsk 10165 df-tskm 10254 |
This theorem is referenced by: eltskm 10259 |
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