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Mirrors > Home > MPE Home > Th. List > elpr | Structured version Visualization version GIF version |
Description: A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elpr.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elpr | ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elprg 4653 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {cpr 4633 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: difprsnss 4804 preq12b 4855 pwpr 4906 pwtp 4907 uniprg 4928 intprg 4986 axprALT 5428 zfpair2 5439 opthwiener 5524 tpres 7221 fnprb 7228 2oconcl 8540 pw2f1olem 9115 djuunxp 9959 sdom2en01 10340 gruun 10844 fzpr 13616 m1expeven 14147 bpoly2 16090 bpoly3 16091 lcmfpr 16661 isprm2 16716 gsumpr 19988 drngnidl 21271 psgninv 21618 psgnodpm 21624 mdetunilem7 22640 indistopon 23024 dfconn2 23443 cnconn 23446 unconn 23453 txindis 23658 txconn 23713 filconn 23907 xpsdsval 24407 rolle 26043 dvivthlem1 26062 ang180lem3 26869 ang180lem4 26870 wilthlem2 27127 sqff1o 27240 ppiub 27263 lgslem1 27356 lgsdir2lem4 27387 lgsdir2lem5 27388 gausslemma2dlem0i 27423 2lgslem3 27463 2lgslem4 27465 nosgnn0 27718 structiedg0val 29054 usgrexmplef 29291 3vfriswmgrlem 30306 prodpr 32833 cycpm2tr 33122 drngmxidlr 33486 lmat22lem 33778 signslema 34556 circlemethhgt 34637 subfacp1lem1 35164 subfacp1lem4 35168 rankeq1o 36153 onsucconni 36420 topdifinfindis 37329 poimirlem9 37616 divrngidl 38015 isfldidl 38055 dihmeetlem2N 41282 wopprc 43019 pw2f1ocnv 43026 kelac2lem 43053 prclaxpr 44948 rnmptpr 45120 cncfiooicclem1 45849 paireqne 47436 31prm 47522 lighneallem4 47535 usgrexmpl2nb1 47927 usgrexmpl2nb2 47928 usgrexmpl2nb4 47930 usgrexmpl2nb5 47931 usgrexmpl2trifr 47932 nn0sumshdiglem2 48472 onsetreclem3 48938 |
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