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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid3 | Structured version Visualization version GIF version |
Description: Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
bj-elid3 | ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-elid 33944 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉))) | |
2 | opelxp 5443 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 2 | anbi1i 614 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ (V × V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉))) |
4 | op1stg 7513 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
5 | op2ndg 7514 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
6 | 4, 5 | eqeq12d 2793 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉) ↔ 𝐴 = 𝐵)) |
7 | 6 | pm5.32i 567 | . . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
8 | simpl 475 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
9 | 8 | anim1i 605 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
10 | simpl 475 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) | |
11 | eleq1 2853 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
12 | 11 | biimpac 471 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
13 | simpr 477 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
14 | 10, 12, 13 | jca31 507 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
15 | 9, 14 | impbii 201 | . . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
16 | 7, 15 | bitri 267 | . . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
17 | 3, 16 | bitri 267 | . 2 ⊢ ((〈𝐴, 𝐵〉 ∈ (V × V) ∧ (1st ‘〈𝐴, 𝐵〉) = (2nd ‘〈𝐴, 𝐵〉)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
18 | 1, 17 | bitri 267 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3415 〈cop 4447 I cid 5311 × cxp 5405 ‘cfv 6188 1st c1st 7499 2nd c2nd 7500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-iota 6152 df-fun 6190 df-fv 6196 df-1st 7501 df-2nd 7502 |
This theorem is referenced by: bj-elid4 33947 bj-eldiag2 33952 |
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