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Mirrors > Home > MPE Home > Th. List > tskmid | Structured version Visualization version GIF version |
Description: The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
Ref | Expression |
---|---|
tskmid | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) | |
2 | 1 | rgenw 3150 | . . 3 ⊢ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥) |
3 | elintrabg 4889 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))) | |
4 | 2, 3 | mpbiri 260 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
5 | tskmval 10261 | . 2 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | |
6 | 4, 5 | eleqtrrd 2916 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∩ cint 4876 ‘cfv 6355 Tarskictsk 10170 tarskiMapctskm 10259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-groth 10245 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-tsk 10171 df-tskm 10260 |
This theorem is referenced by: eltskm 10265 |
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