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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1steq | Structured version Visualization version GIF version |
Description: Uniqueness condition for the binary relation 1st. (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 |
---|---|
br1steq | ⊢ (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1steq.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | br1steq.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | br1steqg 8009 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3469 〈cop 4630 class class class wbr 5142 1st c1st 7985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7987 |
This theorem is referenced by: dfdm5 35304 brtxp 35412 brpprod 35417 elfuns 35447 brimg 35469 brcup 35471 brcap 35472 brrestrict 35481 |
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