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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 2846 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
7 | oprcl 4897 | . . . . . 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 4877 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) |
11 | 10, 5 | eqtr3d 2778 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) |
12 | 8 | simpld 494 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
13 | dfopg 4869 | . . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) | |
14 | 12, 2, 13 | sylancl 586 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) |
15 | 11, 14 | eqtr3d 2778 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐵}}) |
16 | dfopg 4869 | . . . . . . 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 2778 | . . . . 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 4847 | . . . . 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 4732 | . . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷}) | |
24 | 23 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
25 | eqeq2 2748 | . . . . . 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 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
28 | 2, 27 | preqr2 4847 | . . . . 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 4873 | . 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 1540 ∈ wcel 2108 Vcvv 3479 {csn 4624 {cpr 4626 〈cop 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 |
This theorem is referenced by: opthg 5480 otth2 5486 copsexgw 5493 copsexg 5494 copsex2g 5496 copsex4g 5498 opcom 5504 moop2 5505 propssopi 5511 brtp 5526 vopelopabsb 5532 ralxpf 5855 cnvopab 6155 cnvcnvsn 6237 opreu2reurex 6312 funopg 6598 funsndifnop 7169 tpres 7219 f1opr 7487 oprabv 7491 xpopth 8051 eqop 8052 opiota 8080 soxp 8150 fnwelem 8152 xpdom2 9103 xpf1o 9175 unxpdomlem2 9283 unxpdomlem3 9284 xpwdomg 9621 djulf1o 9948 djurf1o 9949 fseqenlem1 10060 iundom2g 10576 eqresr 11173 cnref1o 13023 hashfun 14472 fsumcom2 15806 fprodcom2 16016 qredeu 16691 qnumdenbi 16777 crth 16811 prmreclem3 16952 imasaddfnlem 17569 fnpr2ob 17599 dprd2da 20058 dprd2d2 20060 rngqiprngimf1 21302 ucnima 24280 numclwwlk1lem2f1 30366 brab2d 32608 br8d 32611 xppreima2 32650 aciunf1lem 32661 ofpreima 32664 erdszelem9 35182 goeleq12bg 35332 gonanegoal 35335 gonan0 35375 goaln0 35376 gonarlem 35377 gonar 35378 goalrlem 35379 goalr 35380 fmla0disjsuc 35381 fmlasucdisj 35382 satffunlem 35384 satffunlem1lem1 35385 satffunlem2lem1 35387 msubff1 35539 mvhf1 35542 br8 35734 br6 35735 br4 35736 brsegle 36087 copsex2b 37119 poimirlem4 37609 poimirlem9 37614 dib1dim 41145 diclspsn 41174 dihopelvalcpre 41228 dihmeetlem4preN 41286 dihmeetlem13N 41299 dih1dimatlem 41309 dihatlat 41314 pellexlem3 42820 pellex 42824 snhesn 43777 opelopab4 44549 ichnreuop 47432 ichreuopeq 47433 rrx2xpref1o 48612 brab2dd 48714 |
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