![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opth | Structured version Visualization version GIF version |
Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
opth1.1 | ⊢ 𝐴 ∈ V |
opth1.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opth | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | opth1.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opth1 5478 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) |
4 | 1, 2 | opi1 5471 | . . . . . . 7 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
5 | id 22 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
6 | 4, 5 | eleqtrid 2843 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
7 | oprcl 4906 | . . . . . 6 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
9 | 8 | simprd 495 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐷 ∈ V) |
10 | 3 | opeq1d 4886 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) |
11 | 10, 5 | eqtr3d 2775 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) |
12 | 8 | simpld 494 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
13 | dfopg 4878 | . . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) | |
14 | 12, 2, 13 | sylancl 585 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) |
15 | 11, 14 | eqtr3d 2775 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐵}}) |
16 | dfopg 4878 | . . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐷}}) | |
17 | 8, 16 | syl 17 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐷}}) |
18 | 15, 17 | eqtr3d 2775 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}}) |
19 | prex 5435 | . . . . . 6 ⊢ {𝐶, 𝐵} ∈ V | |
20 | prex 5435 | . . . . . 6 ⊢ {𝐶, 𝐷} ∈ V | |
21 | 19, 20 | preqr2 4856 | . . . . 5 ⊢ ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}) |
22 | 18, 21 | syl 17 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐶, 𝐵} = {𝐶, 𝐷}) |
23 | preq2 4741 | . . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷}) | |
24 | 23 | eqeq2d 2744 | . . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
25 | eqeq2 2745 | . . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐷)) | |
26 | 24, 25 | imbi12d 344 | . . . . 5 ⊢ (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))) |
27 | vex 3481 | . . . . . 6 ⊢ 𝑥 ∈ V | |
28 | 2, 27 | preqr2 4856 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) |
29 | 26, 28 | vtoclg 3553 | . . . 4 ⊢ (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
30 | 9, 22, 29 | sylc 65 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐵 = 𝐷) |
31 | 3, 30 | jca 511 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
32 | opeq12 4882 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
33 | 31, 32 | impbii 209 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 Vcvv 3477 {csn 4630 {cpr 4632 〈cop 4636 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: opthg 5480 otth2 5486 copsexgw 5493 copsexg 5494 copsex2g 5496 copsex4g 5497 opcom 5503 moop2 5504 propssopi 5510 brtp 5525 vopelopabsb 5531 ralxpf 5854 cnvopab 6154 cnvcnvsn 6235 opreu2reurex 6310 funopg 6597 funsndifnop 7165 tpres 7215 f1opr 7483 oprabv 7487 xpopth 8048 eqop 8049 opiota 8077 soxp 8147 fnwelem 8149 xpdom2 9100 xpf1o 9172 unxpdomlem2 9279 unxpdomlem3 9280 xpwdomg 9616 djulf1o 9943 djurf1o 9944 fseqenlem1 10055 iundom2g 10571 eqresr 11168 cnref1o 13018 hashfun 14462 fsumcom2 15796 fprodcom2 16006 qredeu 16681 qnumdenbi 16767 crth 16801 prmreclem3 16941 imasaddfnlem 17564 fnpr2ob 17594 dprd2da 20062 dprd2d2 20064 rngqiprngimf1 21309 ucnima 24287 numclwwlk1lem2f1 30367 brab2d 32607 br8d 32610 xppreima2 32647 aciunf1lem 32658 ofpreima 32661 erdszelem9 35145 goeleq12bg 35295 gonanegoal 35298 gonan0 35338 goaln0 35339 gonarlem 35340 gonar 35341 goalrlem 35342 goalr 35343 fmla0disjsuc 35344 fmlasucdisj 35345 satffunlem 35347 satffunlem1lem1 35348 satffunlem2lem1 35350 msubff1 35502 mvhf1 35505 br8 35696 br6 35697 br4 35698 brsegle 36050 copsex2b 37083 poimirlem4 37571 poimirlem9 37576 dib1dim 41109 diclspsn 41138 dihopelvalcpre 41192 dihmeetlem4preN 41250 dihmeetlem13N 41263 dih1dimatlem 41273 dihatlat 41278 pellexlem3 42773 pellex 42777 snhesn 43734 opelopab4 44508 ichnreuop 47347 ichreuopeq 47348 rrx2xpref1o 48489 |
Copyright terms: Public domain | W3C validator |