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Mirrors > Home > ILE Home > Th. List > en2eleq | GIF version |
Description: Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Ref | Expression |
---|---|
en2eleq | ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6384 | . . . . . . 7 ⊢ 1o ∈ ω | |
2 | simpr 109 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o) | |
3 | df-2o 6282 | . . . . . . . 8 ⊢ 2o = suc 1o | |
4 | 2, 3 | breqtrdi 3939 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ suc 1o) |
5 | simpl 108 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃) | |
6 | dif1en 6741 | . . . . . . 7 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1o) | |
7 | 1, 4, 5, 6 | mp3an2i 1305 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o) |
8 | en1uniel 6666 | . . . . . 6 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋})) | |
9 | 7, 8 | syl 14 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋})) |
10 | eldifsn 3620 | . . . . 5 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)) | |
11 | 9, 10 | sylib 121 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)) |
12 | 11 | simprd 113 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) |
13 | 12 | necomd 2371 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) |
14 | 11 | simpld 111 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃) |
15 | en2eqpr 6769 | . . 3 ⊢ ((𝑃 ≈ 2o ∧ 𝑋 ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃) → (𝑋 ≠ ∪ (𝑃 ∖ {𝑋}) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})) | |
16 | 2, 5, 14, 15 | syl3anc 1201 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑋 ≠ ∪ (𝑃 ∖ {𝑋}) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})) |
17 | 13, 16 | mpd 13 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∖ cdif 3038 {csn 3497 {cpr 3498 ∪ cuni 3706 class class class wbr 3899 suc csuc 4257 ωcom 4474 1oc1o 6274 2oc2o 6275 ≈ cen 6600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-2o 6282 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: en2other2 7020 |
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