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Mirrors > Home > MPE Home > Th. List > Mathboxes > prelspr | Structured version Visualization version GIF version |
Description: An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.) |
Ref | Expression |
---|---|
prelspr | ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prelpwi 5305 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉) | |
2 | eqidd 2799 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌}) | |
3 | preq1 4629 | . . . . . . . 8 ⊢ (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏}) | |
4 | 3 | eqeq2d 2809 | . . . . . . 7 ⊢ (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏})) |
5 | preq2 4630 | . . . . . . . 8 ⊢ (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌}) | |
6 | 5 | eqeq2d 2809 | . . . . . . 7 ⊢ (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌})) |
7 | 4, 6 | rspc2ev 3583 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}) |
8 | 2, 7 | mpd3an3 1459 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}) |
9 | 1, 8 | jca 515 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
11 | eqeq1 2802 | . . . . 5 ⊢ (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏})) | |
12 | 11 | 2rexbidv 3259 | . . . 4 ⊢ (𝑝 = {𝑋, 𝑌} → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
13 | 12 | elrab 3628 | . . 3 ⊢ ({𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ↔ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
14 | 10, 13 | sylibr 237 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) |
15 | sprvalpw 43997 | . . 3 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | |
16 | 15 | adantr 484 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) |
17 | 14, 16 | eleqtrrd 2893 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 𝒫 cpw 4497 {cpr 4527 ‘cfv 6324 Pairscspr 43994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-spr 43995 |
This theorem is referenced by: sprsymrelfolem2 44010 reupr 44039 |
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