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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 5330 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉) | |
2 | eqidd 2819 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌}) | |
3 | preq1 4661 | . . . . . . . 8 ⊢ (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏}) | |
4 | 3 | eqeq2d 2829 | . . . . . . 7 ⊢ (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏})) |
5 | preq2 4662 | . . . . . . . 8 ⊢ (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌}) | |
6 | 5 | eqeq2d 2829 | . . . . . . 7 ⊢ (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌})) |
7 | 4, 6 | rspc2ev 3632 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}) |
8 | 2, 7 | mpd3an3 1453 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}) |
9 | 1, 8 | jca 512 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
10 | 9 | adantl 482 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
11 | eqeq1 2822 | . . . . 5 ⊢ (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏})) | |
12 | 11 | 2rexbidv 3297 | . . . 4 ⊢ (𝑝 = {𝑋, 𝑌} → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})) |
13 | 12 | elrab 3677 | . . 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 235 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) |
15 | sprvalpw 43519 | . . 3 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | |
16 | 15 | adantr 481 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) |
17 | 14, 16 | eleqtrrd 2913 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 {crab 3139 𝒫 cpw 4535 {cpr 4559 ‘cfv 6348 Pairscspr 43516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-spr 43517 |
This theorem is referenced by: sprsymrelfolem2 43532 reupr 43561 |
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