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| Mirrors > Home > MPE Home > Th. List > prfi | Structured version Visualization version GIF version | ||
| Description: An unordered pair is finite. For a shorter proof using ax-un 7755, see prfiALT 9364. (Contributed by NM, 22-Aug-2008.) Avoid ax-11 2157, ax-un 7755. (Revised by BTernaryTau, 13-Jan-2025.) |
| Ref | Expression |
|---|---|
| prfi | ⊢ {𝐴, 𝐵} ∈ Fin |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prprc1 4765 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
| 2 | snfi 9083 | . . 3 ⊢ {𝐵} ∈ Fin | |
| 3 | 1, 2 | eqeltrdi 2849 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} ∈ Fin) |
| 4 | prprc2 4766 | . . 3 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) | |
| 5 | snfi 9083 | . . 3 ⊢ {𝐴} ∈ Fin | |
| 6 | 4, 5 | eqeltrdi 2849 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} ∈ Fin) |
| 7 | 2onn 8680 | . . . . . 6 ⊢ 2o ∈ ω | |
| 8 | simp1 1137 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ V) | |
| 9 | simp2 1138 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ V) | |
| 10 | simp3 1139 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
| 11 | 8, 9, 10 | enpr2d 9089 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ≈ 2o) |
| 12 | breq2 5147 | . . . . . . 7 ⊢ (𝑥 = 2o → ({𝐴, 𝐵} ≈ 𝑥 ↔ {𝐴, 𝐵} ≈ 2o)) | |
| 13 | 12 | rspcev 3622 | . . . . . 6 ⊢ ((2o ∈ ω ∧ {𝐴, 𝐵} ≈ 2o) → ∃𝑥 ∈ ω {𝐴, 𝐵} ≈ 𝑥) |
| 14 | 7, 11, 13 | sylancr 587 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → ∃𝑥 ∈ ω {𝐴, 𝐵} ≈ 𝑥) |
| 15 | isfi 9016 | . . . . 5 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴, 𝐵} ≈ 𝑥) | |
| 16 | 14, 15 | sylibr 234 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ∈ Fin) |
| 17 | 16 | 3expia 1122 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ∈ Fin)) |
| 18 | dfsn2 4639 | . . . . 5 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 19 | preq2 4734 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
| 20 | 18, 19 | eqtr2id 2790 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
| 21 | 20, 5 | eqeltrdi 2849 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} ∈ Fin) |
| 22 | 17, 21 | pm2.61d2 181 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ Fin) |
| 23 | 3, 6, 22 | ecase 1034 | 1 ⊢ {𝐴, 𝐵} ∈ Fin |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 {csn 4626 {cpr 4628 class class class wbr 5143 ωcom 7887 2oc2o 8500 ≈ cen 8982 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-1o 8506 df-2o 8507 df-en 8986 df-fin 8989 |
| This theorem is referenced by: tpfi 9365 fiint 9366 fiintOLD 9367 inelfi 9458 tskpr 10810 hashpw 14475 hashfun 14476 pr2pwpr 14518 hashtpg 14524 hash3tpexb 14533 sumpr 15784 lcmfpr 16664 prmreclem2 16955 acsfn2 17706 isdrs2 18352 efmnd2hash 18907 symg2hash 19409 psgnprfval 19539 gsumpr 19973 znidomb 21580 m2detleib 22637 ovolioo 25603 i1f1 25725 itgioo 25851 limcun 25930 aannenlem2 26371 wilthlem2 27112 perfectlem2 27274 upgrex 29109 ex-hash 30472 prodpr 32828 linds2eq 33409 elrspunsn 33457 constrfin 33787 inelpisys 34155 coinfliplem 34481 coinflippv 34486 subfacp1lem1 35184 poimirlem9 37636 kelac2lem 43076 sumpair 45040 refsum2cnlem1 45042 climxlim2lem 45860 ibliooicc 45986 fourierdlem50 46171 fourierdlem51 46172 fourierdlem54 46175 fourierdlem70 46191 fourierdlem71 46192 fourierdlem76 46197 fourierdlem102 46223 fourierdlem103 46224 fourierdlem104 46225 fourierdlem114 46235 saluncl 46332 sge0pr 46409 meadjun 46477 omeunle 46531 perfectALTVlem2 47709 gpgorder 48013 zlmodzxzel 48271 ldepspr 48390 zlmodzxzldeplem2 48418 rrx2line 48661 2sphere 48670 |
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