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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 7712 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 2nd c2nd 7688 |
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 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-2nd 7690 |
This theorem is referenced by: dfrn5 33017 brtxp 33341 brpprod 33346 elfuns 33376 brimg 33398 brcup 33400 brcap 33401 brrestrict 33410 |
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