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Mirrors > Home > MPE Home > Th. List > en2other2 | Structured version Visualization version 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 10033 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) | |
2 | prcom 4738 | . . . . . . 7 ⊢ {𝑋, ∪ (𝑃 ∖ {𝑋})} = {∪ (𝑃 ∖ {𝑋}), 𝑋} | |
3 | 1, 2 | eqtrdi 2781 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {∪ (𝑃 ∖ {𝑋}), 𝑋}) |
4 | 3 | difeq1d 4117 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})})) |
5 | difprsnss 4804 | . . . . 5 ⊢ ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋} | |
6 | 4, 5 | eqsstrdi 4031 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋}) |
7 | simpl 481 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃) | |
8 | 1onn 8661 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
9 | simpr 483 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o) | |
10 | df-2o 8488 | . . . . . . . . . 10 ⊢ 2o = suc 1o | |
11 | 9, 10 | breqtrdi 5190 | . . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ suc 1o) |
12 | dif1ennn 9186 | . . . . . . . . 9 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1o) | |
13 | 8, 11, 7, 12 | mp3an2i 1462 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o) |
14 | en1uniel 9053 | . . . . . . . 8 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋})) | |
15 | eldifsni 4795 | . . . . . . . 8 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) | |
16 | 13, 14, 15 | 3syl 18 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) |
17 | 16 | necomd 2985 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) |
18 | eldifsn 4792 | . . . . . 6 ⊢ (𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ↔ (𝑋 ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))) | |
19 | 7, 17, 18 | sylanbrc 581 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})})) |
20 | 19 | snssd 4814 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋} ⊆ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})})) |
21 | 6, 20 | eqssd 3994 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = {𝑋}) |
22 | 21 | unieqd 4922 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ∪ {𝑋}) |
23 | unisng 4929 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ∪ {𝑋} = 𝑋) | |
24 | 23 | adantr 479 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ {𝑋} = 𝑋) |
25 | 22, 24 | eqtrd 2765 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3941 {csn 4630 {cpr 4632 ∪ cuni 4909 class class class wbr 5149 suc csuc 6373 ωcom 7871 1oc1o 8480 2oc2o 8481 ≈ cen 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 |
This theorem is referenced by: pmtrfinv 19428 |
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