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Mirrors > Home > MPE Home > Th. List > fisn | Structured version Visualization version GIF version |
Description: A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
fisn | ⊢ (fi‘{𝐴}) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4650 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | elsni 4650 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
3 | 1, 2 | ineqan12d 4215 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴)) |
4 | inidm 4220 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | 3, 4 | eqtrdi 2782 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) = 𝐴) |
6 | vex 3466 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | inex1 5322 | . . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ V |
8 | 7 | elsn 4648 | . . . 4 ⊢ ((𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (𝑥 ∩ 𝑦) = 𝐴) |
9 | 5, 8 | sylibr 233 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) ∈ {𝐴}) |
10 | 9 | rgen2 3188 | . 2 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} |
11 | snex 5437 | . . 3 ⊢ {𝐴} ∈ V | |
12 | inficl 9468 | . . 3 ⊢ ({𝐴} ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (fi‘{𝐴}) = {𝐴})) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (fi‘{𝐴}) = {𝐴}) |
14 | 10, 13 | mpbi 229 | 1 ⊢ (fi‘{𝐴}) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∩ cin 3946 {csn 4633 ‘cfv 6554 ficfi 9453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-om 7877 df-1o 8496 df-2o 8497 df-en 8975 df-fin 8978 df-fi 9454 |
This theorem is referenced by: ordtbas 23187 |
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