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Mirrors > Home > NFE Home > Th. List > opbr1st | GIF version |
Description: Binary relationship of an ordered pair over 1st. (Contributed by SF, 6-Feb-2015.) |
Ref | Expression |
---|---|
opbr1st.1 | ⊢ A ∈ V |
opbr1st.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
opbr1st | ⊢ (〈A, B〉1st C ↔ A = C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 ⊢ (〈A, B〉1st C → (〈A, B〉 ∈ V ∧ C ∈ V)) | |
2 | 1 | simprd 449 | . 2 ⊢ (〈A, B〉1st C → C ∈ V) |
3 | opbr1st.1 | . . 3 ⊢ A ∈ V | |
4 | eleq1 2413 | . . 3 ⊢ (A = C → (A ∈ V ↔ C ∈ V)) | |
5 | 3, 4 | mpbii 202 | . 2 ⊢ (A = C → C ∈ V) |
6 | breq2 4643 | . . 3 ⊢ (x = C → (〈A, B〉1st x ↔ 〈A, B〉1st C)) | |
7 | eqeq2 2362 | . . 3 ⊢ (x = C → (A = x ↔ A = C)) | |
8 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
9 | 8 | br1st 4858 | . . . 4 ⊢ (〈A, B〉1st x ↔ ∃y〈A, B〉 = 〈x, y〉) |
10 | opbr1st.2 | . . . . . 6 ⊢ B ∈ V | |
11 | biidd 228 | . . . . . 6 ⊢ (y = B → (x = A ↔ x = A)) | |
12 | 10, 11 | ceqsexv 2894 | . . . . 5 ⊢ (∃y(y = B ∧ x = A) ↔ x = A) |
13 | eqcom 2355 | . . . . . . 7 ⊢ (〈A, B〉 = 〈x, y〉 ↔ 〈x, y〉 = 〈A, B〉) | |
14 | opth 4602 | . . . . . . . 8 ⊢ (〈x, y〉 = 〈A, B〉 ↔ (x = A ∧ y = B)) | |
15 | ancom 437 | . . . . . . . 8 ⊢ ((x = A ∧ y = B) ↔ (y = B ∧ x = A)) | |
16 | 14, 15 | bitri 240 | . . . . . . 7 ⊢ (〈x, y〉 = 〈A, B〉 ↔ (y = B ∧ x = A)) |
17 | 13, 16 | bitri 240 | . . . . . 6 ⊢ (〈A, B〉 = 〈x, y〉 ↔ (y = B ∧ x = A)) |
18 | 17 | exbii 1582 | . . . . 5 ⊢ (∃y〈A, B〉 = 〈x, y〉 ↔ ∃y(y = B ∧ x = A)) |
19 | eqcom 2355 | . . . . 5 ⊢ (A = x ↔ x = A) | |
20 | 12, 18, 19 | 3bitr4i 268 | . . . 4 ⊢ (∃y〈A, B〉 = 〈x, y〉 ↔ A = x) |
21 | 9, 20 | bitri 240 | . . 3 ⊢ (〈A, B〉1st x ↔ A = x) |
22 | 6, 7, 21 | vtoclbg 2915 | . 2 ⊢ (C ∈ V → (〈A, B〉1st C ↔ A = C)) |
23 | 2, 5, 22 | pm5.21nii 342 | 1 ⊢ (〈A, B〉1st C ↔ A = C) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 〈cop 4561 class class class wbr 4639 1st c1st 4717 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 |
This theorem is referenced by: 1stfo 5505 opfv1st 5514 brco1st 5777 trtxp 5781 op1st2nd 5790 oqelins4 5794 qrpprod 5836 xpassenlem 6056 xpassen 6057 enpw1lem1 6061 nncdiv3lem1 6275 |
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