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Mirrors > Home > ILE Home > Th. List > en2other2 | GIF version |
Description: Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Ref | Expression |
---|---|
en2other2 | ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2eleq 6573 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) | |
2 | prcom 3486 | . . . . . . 7 ⊢ {𝑋, ∪ (𝑃 ∖ {𝑋})} = {∪ (𝑃 ∖ {𝑋}), 𝑋} | |
3 | 1, 2 | syl6eq 2131 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {∪ (𝑃 ∖ {𝑋}), 𝑋}) |
4 | 3 | difeq1d 3099 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})})) |
5 | difprsnss 3543 | . . . . 5 ⊢ ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋} | |
6 | 4, 5 | syl6eqss 3058 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋}) |
7 | simpl 107 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ 𝑃) | |
8 | 1onn 6180 | . . . . . . . . . 10 ⊢ 1𝑜 ∈ ω | |
9 | 8 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω) |
10 | simpr 108 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜) | |
11 | df-2o 6086 | . . . . . . . . . 10 ⊢ 2𝑜 = suc 1𝑜 | |
12 | 10, 11 | syl6breq 3844 | . . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜) |
13 | dif1en 6435 | . . . . . . . . 9 ⊢ ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜 ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜) | |
14 | 9, 12, 7, 13 | syl3anc 1170 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜) |
15 | en1uniel 6372 | . . . . . . . 8 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1𝑜 → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋})) | |
16 | eldifsni 3537 | . . . . . . . 8 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) | |
17 | 14, 15, 16 | 3syl 17 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) |
18 | 17 | necomd 2335 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) |
19 | eldifsn 3535 | . . . . . 6 ⊢ (𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ↔ (𝑋 ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))) | |
20 | 7, 18, 19 | sylanbrc 408 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})})) |
21 | 20 | snssd 3550 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋} ⊆ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})})) |
22 | 6, 21 | eqssd 3025 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = {𝑋}) |
23 | 22 | unieqd 3632 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ∪ {𝑋}) |
24 | unisng 3638 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ∪ {𝑋} = 𝑋) | |
25 | 24 | adantr 270 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ {𝑋} = 𝑋) |
26 | 23, 25 | eqtrd 2115 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 ∖ cdif 2979 {csn 3416 {cpr 3417 ∪ cuni 3621 class class class wbr 3805 suc csuc 4148 ωcom 4359 1𝑜c1o 6078 2𝑜c2o 6079 ≈ cen 6306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-if 3369 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-1o 6085 df-2o 6086 df-er 6193 df-en 6309 df-fin 6311 |
This theorem is referenced by: (None) |
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