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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 10774 | . . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | |
2 | ssrab2 4037 | . . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski | |
3 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
4 | grothtsk 10770 | . . . . . . 7 ⊢ ∪ Tarski = V | |
5 | 3, 4 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski) |
6 | eluni2 4869 | . . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
7 | 5, 6 | sylib 217 | . . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) |
8 | rabn0 4345 | . . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
9 | 7, 8 | sylibr 233 | . . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) |
10 | inttsk 10709 | . . . 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 2838 | . 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski) |
13 | fvprc 6834 | . . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅) | |
14 | 0tsk 10690 | . . 3 ⊢ ∅ ∈ Tarski | |
15 | 13, 14 | eqeltrdi 2846 | . 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 2106 ≠ wne 2943 ∃wrex 3073 {crab 3407 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ∪ cuni 4865 ∩ cint 4907 ‘cfv 6496 Tarskictsk 10683 tarskiMapctskm 10772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-groth 10758 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-er 8647 df-en 8883 df-dom 8884 df-tsk 10684 df-tskm 10773 |
This theorem is referenced by: eltskm 10778 |
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