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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 7753, see prfiALT 9361. (Contributed by NM, 22-Aug-2008.) Avoid ax-11 2154, ax-un 7753. (Revised by BTernaryTau, 13-Jan-2025.) |
Ref | Expression |
---|---|
prfi | ⊢ {𝐴, 𝐵} ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prprc1 4769 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | |
2 | snfi 9081 | . . 3 ⊢ {𝐵} ∈ Fin | |
3 | 1, 2 | eqeltrdi 2846 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} ∈ Fin) |
4 | prprc2 4770 | . . 3 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) | |
5 | snfi 9081 | . . 3 ⊢ {𝐴} ∈ Fin | |
6 | 4, 5 | eqeltrdi 2846 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} ∈ Fin) |
7 | 2onn 8678 | . . . . . 6 ⊢ 2o ∈ ω | |
8 | simp1 1135 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ V) | |
9 | simp2 1136 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ V) | |
10 | simp3 1137 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
11 | 8, 9, 10 | enpr2d 9087 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ≈ 2o) |
12 | breq2 5151 | . . . . . . 7 ⊢ (𝑥 = 2o → ({𝐴, 𝐵} ≈ 𝑥 ↔ {𝐴, 𝐵} ≈ 2o)) | |
13 | 12 | rspcev 3621 | . . . . . 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 9014 | . . . . 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 1120 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (¬ 𝐴 = 𝐵 → {𝐴, 𝐵} ∈ Fin)) |
18 | dfsn2 4643 | . . . . 5 ⊢ {𝐴} = {𝐴, 𝐴} | |
19 | preq2 4738 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
20 | 18, 19 | eqtr2id 2787 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
21 | 20, 5 | eqeltrdi 2846 | . . 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 1033 | 1 ⊢ {𝐴, 𝐵} ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 {csn 4630 {cpr 4632 class class class wbr 5147 ωcom 7886 2oc2o 8498 ≈ cen 8980 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-om 7887 df-1o 8504 df-2o 8505 df-en 8984 df-fin 8987 |
This theorem is referenced by: tpfi 9362 fiint 9363 fiintOLD 9364 inelfi 9455 tskpr 10807 hashpw 14471 hashfun 14472 pr2pwpr 14514 hashtpg 14520 hash3tpexb 14529 sumpr 15780 lcmfpr 16660 prmreclem2 16950 acsfn2 17707 isdrs2 18363 efmnd2hash 18919 symg2hash 19423 psgnprfval 19553 gsumpr 19987 znidomb 21597 m2detleib 22652 ovolioo 25616 i1f1 25738 itgioo 25865 limcun 25944 aannenlem2 26385 wilthlem2 27126 perfectlem2 27288 upgrex 29123 ex-hash 30481 prodpr 32832 linds2eq 33388 elrspunsn 33436 constrfin 33750 inelpisys 34134 coinfliplem 34459 coinflippv 34464 subfacp1lem1 35163 poimirlem9 37615 kelac2lem 43052 sumpair 44972 refsum2cnlem1 44974 climxlim2lem 45800 ibliooicc 45926 fourierdlem50 46111 fourierdlem51 46112 fourierdlem54 46115 fourierdlem70 46131 fourierdlem71 46132 fourierdlem76 46137 fourierdlem102 46163 fourierdlem103 46164 fourierdlem104 46165 fourierdlem114 46175 saluncl 46272 sge0pr 46349 meadjun 46417 omeunle 46471 perfectALTVlem2 47646 gpgorder 47947 zlmodzxzel 48199 ldepspr 48318 zlmodzxzldeplem2 48346 rrx2line 48589 2sphere 48598 |
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