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Mirrors > Home > MPE Home > Th. List > infex | Structured version Visualization version GIF version |
Description: An infimum is a set. (Contributed by AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infex.1 | ⊢ 𝑅 Or 𝐴 |
Ref | Expression |
---|---|
infex | ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infex.1 | . 2 ⊢ 𝑅 Or 𝐴 | |
2 | id 22 | . . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Or 𝐴) | |
3 | 2 | infexd 9552 | . 2 ⊢ (𝑅 Or 𝐴 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 Vcvv 3488 Or wor 5606 infcinf 9510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-po 5607 df-so 5608 df-cnv 5708 df-sup 9511 df-inf 9512 |
This theorem is referenced by: limsupval 15520 lcmval 16639 odzval 16838 ramval 17055 imasdsfn 17574 imasdsval 17575 odval 19576 odf 19579 gexval 19620 nmoval 24757 metdsval 24888 ovolval 25527 ovolf 25536 elqaalem1 26379 elqaalem3 26381 ballotlemi 34465 pellfundval 42836 dgraaval 43101 dgraaf 43104 liminfgval 45683 liminfval2 45689 ovnval2 46466 finfdm2 46768 |
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