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Mirrors > Home > MPE Home > Th. List > Mathboxes > prprelb | Structured version Visualization version GIF version |
Description: An element of the set of all proper unordered pairs over a given set 𝑉 is a subset of 𝑉 of size two. (Contributed by AV, 29-Apr-2023.) |
Ref | Expression |
---|---|
prprelb | ⊢ (𝑉 ∈ 𝑊 → (𝑃 ∈ (Pairsproper‘𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prprvalpw 43726 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}) | |
2 | 1 | eleq2d 2898 | . . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑃 ∈ (Pairsproper‘𝑉) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})) |
3 | eqeq1 2825 | . . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 = {𝑎, 𝑏} ↔ 𝑃 = {𝑎, 𝑏})) | |
4 | 3 | anbi2d 630 | . . . . 5 ⊢ (𝑝 = 𝑃 → ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) |
5 | 4 | 2rexbidv 3300 | . . . 4 ⊢ (𝑝 = 𝑃 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) |
6 | 5 | elrab 3680 | . . 3 ⊢ (𝑃 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ↔ (𝑃 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) |
7 | 2, 6 | syl6bb 289 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑃 ∈ (Pairsproper‘𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))) |
8 | hash2exprb 13830 | . . . 4 ⊢ (𝑃 ∈ 𝒫 𝑉 → ((♯‘𝑃) = 2 ↔ ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) | |
9 | eleq1 2900 | . . . . . . . . . 10 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉)) | |
10 | prelpw 5339 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉)) | |
11 | 10 | el2v 3501 | . . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉) |
12 | 11 | biimpri 230 | . . . . . . . . . 10 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
13 | 9, 12 | syl6bi 255 | . . . . . . . . 9 ⊢ (𝑃 = {𝑎, 𝑏} → (𝑃 ∈ 𝒫 𝑉 → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
14 | 13 | com12 32 | . . . . . . . 8 ⊢ (𝑃 ∈ 𝒫 𝑉 → (𝑃 = {𝑎, 𝑏} → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
15 | 14 | adantld 493 | . . . . . . 7 ⊢ (𝑃 ∈ 𝒫 𝑉 → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
16 | 15 | pm4.71rd 565 | . . . . . 6 ⊢ (𝑃 ∈ 𝒫 𝑉 → ((𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))) |
17 | 16 | 2exbidv 1925 | . . . . 5 ⊢ (𝑃 ∈ 𝒫 𝑉 → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})))) |
18 | r2ex 3303 | . . . . 5 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) | |
19 | 17, 18 | syl6bbr 291 | . . . 4 ⊢ (𝑃 ∈ 𝒫 𝑉 → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}))) |
20 | 8, 19 | bitr2d 282 | . . 3 ⊢ (𝑃 ∈ 𝒫 𝑉 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏}) ↔ (♯‘𝑃) = 2)) |
21 | 20 | pm5.32i 577 | . 2 ⊢ ((𝑃 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑃 = {𝑎, 𝑏})) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2)) |
22 | 7, 21 | syl6bb 289 | 1 ⊢ (𝑉 ∈ 𝑊 → (𝑃 ∈ (Pairsproper‘𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 𝒫 cpw 4539 {cpr 4569 ‘cfv 6355 2c2 11693 ♯chash 13691 Pairspropercprpr 43723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-prpr 43724 |
This theorem is referenced by: prprreueq 43731 |
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