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Mirrors > Home > MPE Home > Th. List > Mathboxes > prpair | Structured version Visualization version GIF version |
Description: Characterization of a proper pair: A class is a proper pair iff it consists of exactly two different sets. (Contributed by AV, 11-Mar-2023.) |
Ref | Expression |
---|---|
prpair.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Ref | Expression |
---|---|
prpair | ⊢ (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prpair.p | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | |
2 | 1 | eleq2i 2818 | . 2 ⊢ (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
3 | fveqeq2 6910 | . . 3 ⊢ (𝑥 = 𝑋 → ((♯‘𝑥) = 2 ↔ (♯‘𝑋) = 2)) | |
4 | 3 | elrab 3681 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2)) |
5 | hash2prb 14491 | . . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 → ((♯‘𝑋) = 2 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑎, 𝑏}))) | |
6 | elpwi 4614 | . . . . . 6 ⊢ (𝑋 ∈ 𝒫 𝑉 → 𝑋 ⊆ 𝑉) | |
7 | ancom 459 | . . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑎, 𝑏}) ↔ (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) | |
8 | 7 | 2rexbii 3119 | . . . . . . 7 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
9 | 8 | biimpi 215 | . . . . . 6 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑎, 𝑏}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
10 | ss2rexv 4051 | . . . . . 6 ⊢ (𝑋 ⊆ 𝑉 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))) | |
11 | 6, 9, 10 | syl2im 40 | . . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑉 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑎, 𝑏}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))) |
12 | 5, 11 | sylbid 239 | . . . 4 ⊢ (𝑋 ∈ 𝒫 𝑉 → ((♯‘𝑋) = 2 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))) |
13 | 12 | imp 405 | . . 3 ⊢ ((𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
14 | prelpwi 5453 | . . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → {𝑎, 𝑏} ∈ 𝒫 𝑉) | |
15 | 14 | adantr 479 | . . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → {𝑎, 𝑏} ∈ 𝒫 𝑉) |
16 | hashprg 14412 | . . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 ↔ (♯‘{𝑎, 𝑏}) = 2)) | |
17 | 16 | biimpd 228 | . . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 → (♯‘{𝑎, 𝑏}) = 2)) |
18 | 17 | adantld 489 | . . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (♯‘{𝑎, 𝑏}) = 2)) |
19 | 18 | imp 405 | . . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (♯‘{𝑎, 𝑏}) = 2) |
20 | eleq1 2814 | . . . . . . . . 9 ⊢ (𝑋 = {𝑎, 𝑏} → (𝑋 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉)) | |
21 | fveqeq2 6910 | . . . . . . . . 9 ⊢ (𝑋 = {𝑎, 𝑏} → ((♯‘𝑋) = 2 ↔ (♯‘{𝑎, 𝑏}) = 2)) | |
22 | 20, 21 | anbi12d 630 | . . . . . . . 8 ⊢ (𝑋 = {𝑎, 𝑏} → ((𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2) ↔ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ∧ (♯‘{𝑎, 𝑏}) = 2))) |
23 | 22 | adantr 479 | . . . . . . 7 ⊢ ((𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2) ↔ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ∧ (♯‘{𝑎, 𝑏}) = 2))) |
24 | 23 | adantl 480 | . . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → ((𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2) ↔ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ∧ (♯‘{𝑎, 𝑏}) = 2))) |
25 | 15, 19, 24 | mpbir2and 711 | . . . . 5 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2)) |
26 | 25 | ex 411 | . . . 4 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2))) |
27 | 26 | rexlimivv 3190 | . . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2)) |
28 | 13, 27 | impbii 208 | . 2 ⊢ ((𝑋 ∈ 𝒫 𝑉 ∧ (♯‘𝑋) = 2) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
29 | 2, 4, 28 | 3bitri 296 | 1 ⊢ (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 {crab 3419 ⊆ wss 3947 𝒫 cpw 4607 {cpr 4635 ‘cfv 6554 2c2 12319 ♯chash 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-hash 14348 |
This theorem is referenced by: prproropf1olem2 47076 prproropf1olem4 47078 |
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