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Mirrors > Home > MPE Home > Th. List > Mathboxes > br2ndeq | Structured version Visualization version GIF version |
Description: Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
Ref | Expression |
---|---|
br1steq.1 | ⊢ 𝐴 ∈ V |
br1steq.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
br2ndeq | ⊢ (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1steq.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | br1steq.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | br2ndeqg 7762 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 class class class wbr 5039 2nd c2nd 7738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-2nd 7740 |
This theorem is referenced by: dfrn5 33418 brtxp 33868 brpprod 33873 elfuns 33903 brimg 33925 brcup 33927 brcap 33928 brrestrict 33937 |
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