Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > prid1g | Structured version Visualization version GIF version |
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
Ref | Expression |
---|---|
prid1g | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | orci 861 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵) |
3 | elprg 4588 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵))) | |
4 | 2, 3 | mpbiri 260 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-sn 4568 df-pr 4570 |
This theorem is referenced by: prid2g 4697 prid1 4698 prnzg 4713 preq1b 4777 prel12g 4794 elpreqprb 4798 prproe 4836 opth1 5367 fr2nr 5533 fpr2g 6974 f1prex 7040 fveqf1o 7058 pw2f1olem 8621 hashprdifel 13760 gcdcllem3 15850 mgm2nsgrplem1 18083 mgm2nsgrplem2 18084 mgm2nsgrplem3 18085 sgrp2nmndlem1 18088 sgrp2rid2 18091 pmtrprfv 18581 pptbas 21616 coseq0negpitopi 25089 uhgr2edg 26990 umgrvad2edg 26995 uspgr2v1e2w 27033 usgr2v1e2w 27034 nbusgredgeu0 27150 nbusgrf1o0 27151 nb3grprlem1 27162 nb3grprlem2 27163 vtxduhgr0nedg 27274 1hegrvtxdg1 27289 1egrvtxdg1 27291 umgr2v2evd2 27309 vdegp1bi 27319 mptprop 30434 altgnsg 30791 cyc3genpmlem 30793 bj-prmoore 34410 ftc1anclem8 34989 kelac2 39685 pr2el1 39928 pr2eldif1 39933 fourierdlem54 42465 sge0pr 42696 imarnf1pr 43501 paireqne 43693 fmtnoprmfac2lem1 43748 1hegrlfgr 44027 |
Copyright terms: Public domain | W3C validator |