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Mirrors > Home > MPE Home > Th. List > exprelprel | Structured version Visualization version GIF version |
Description: If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.) |
Ref | Expression |
---|---|
exprelprel | ⊢ (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elss2prb 13839 | . . 3 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 (𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤})) | |
2 | eleq1 2900 | . . . . . . . 8 ⊢ (𝑝 = {𝑣, 𝑤} → (𝑝 ∈ 𝑋 ↔ {𝑣, 𝑤} ∈ 𝑋)) | |
3 | 2 | adantl 484 | . . . . . . 7 ⊢ ((𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤}) → (𝑝 ∈ 𝑋 ↔ {𝑣, 𝑤} ∈ 𝑋)) |
4 | 3 | biimpcd 251 | . . . . . 6 ⊢ (𝑝 ∈ 𝑋 → ((𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤}) → {𝑣, 𝑤} ∈ 𝑋)) |
5 | 4 | reximdv 3273 | . . . . 5 ⊢ (𝑝 ∈ 𝑋 → (∃𝑤 ∈ 𝑉 (𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤}) → ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)) |
6 | 5 | reximdv 3273 | . . . 4 ⊢ (𝑝 ∈ 𝑋 → (∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 (𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤}) → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)) |
7 | 6 | com12 32 | . . 3 ⊢ (∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 (𝑣 ≠ 𝑤 ∧ 𝑝 = {𝑣, 𝑤}) → (𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)) |
8 | 1, 7 | sylbi 219 | . 2 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} → (𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)) |
9 | 8 | rexlimiv 3280 | 1 ⊢ (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 𝒫 cpw 4538 {cpr 4562 ‘cfv 6349 2c2 11686 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 |
This theorem is referenced by: (None) |
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