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Mirrors > Home > ILE Home > Th. List > undiffi | GIF version |
Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3382 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.) |
Ref | Expression |
---|---|
undiffi | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidceq 6665 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → DECID 𝑥 = 𝑦) | |
2 | 1 | 3expb 1147 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID 𝑥 = 𝑦) |
3 | 2 | ralrimivva 2467 | . 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
4 | undifdc 6714 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) | |
5 | 3, 4 | syl3an1 1214 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 783 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ∖ cdif 3010 ∪ cun 3011 ⊆ wss 3013 Fincfn 6537 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-er 6332 df-en 6538 df-fin 6540 |
This theorem is referenced by: unfiin 6716 fihashssdif 10357 fsumlessfi 11018 |
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