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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 7994 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 class class class wbr 5141 2nd c2nd 7970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-2nd 7972 |
This theorem is referenced by: dfrn5 35278 brtxp 35385 brpprod 35390 elfuns 35420 brimg 35442 brcup 35444 brcap 35445 brrestrict 35454 |
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