![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wuncidm | Structured version Visualization version GIF version |
Description: The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wuncidm | ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunccl 9882 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni) | |
2 | ssid 3849 | . . 3 ⊢ (wUniCl‘𝐴) ⊆ (wUniCl‘𝐴) | |
3 | wuncss 9883 | . . 3 ⊢ (((wUniCl‘𝐴) ∈ WUni ∧ (wUniCl‘𝐴) ⊆ (wUniCl‘𝐴)) → (wUniCl‘(wUniCl‘𝐴)) ⊆ (wUniCl‘𝐴)) | |
4 | 1, 2, 3 | sylancl 582 | . 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) ⊆ (wUniCl‘𝐴)) |
5 | wuncid 9881 | . . 3 ⊢ ((wUniCl‘𝐴) ∈ WUni → (wUniCl‘𝐴) ⊆ (wUniCl‘(wUniCl‘𝐴))) | |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ⊆ (wUniCl‘(wUniCl‘𝐴))) |
7 | 4, 6 | eqssd 3845 | 1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3799 ‘cfv 6124 WUnicwun 9838 wUniClcwunm 9839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-wun 9840 df-wunc 9841 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |