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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 5071 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) |
4 | 1, 2 | opi1 5064 | . . . . . . 7 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
5 | id 22 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
6 | 4, 5 | syl5eleq 2856 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
7 | oprcl 4565 | . . . . . 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 483 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐷 ∈ V) |
10 | 3 | opeq1d 4545 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) |
11 | 10, 5 | eqtr3d 2807 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) |
12 | 8 | simpld 482 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
13 | dfopg 4537 | . . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) | |
14 | 12, 2, 13 | sylancl 574 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐵〉 = {{𝐶}, {𝐶, 𝐵}}) |
15 | 11, 14 | eqtr3d 2807 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐵}}) |
16 | dfopg 4537 | . . . . . . 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 2807 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}}) |
19 | prex 5037 | . . . . . 6 ⊢ {𝐶, 𝐵} ∈ V | |
20 | prex 5037 | . . . . . 6 ⊢ {𝐶, 𝐷} ∈ V | |
21 | 19, 20 | preqr2 4512 | . . . . 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 4405 | . . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷}) | |
24 | 23 | eqeq2d 2781 | . . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
25 | eqeq2 2782 | . . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐷)) | |
26 | 24, 25 | imbi12d 333 | . . . . 5 ⊢ (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))) |
27 | vex 3354 | . . . . . 6 ⊢ 𝑥 ∈ V | |
28 | 2, 27 | preqr2 4512 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) |
29 | 26, 28 | vtoclg 3417 | . . . 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 501 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
32 | opeq12 4541 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
33 | 31, 32 | impbii 199 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 {csn 4316 {cpr 4318 〈cop 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 |
This theorem is referenced by: opthg 5073 otth2 5079 copsexg 5083 copsex4g 5086 opcom 5092 moop2 5093 propssopi 5101 opelopabsbALT 5117 ralxpf 5407 cnvcnvsn 5754 funopg 6065 funsndifnop 6559 tpres 6610 oprabv 6850 xpopth 7356 eqop 7357 opiota 7378 soxp 7441 fnwelem 7443 xpdom2 8211 xpf1o 8278 unxpdomlem2 8321 unxpdomlem3 8322 xpwdomg 8646 djulf1o 8938 djurf1o 8939 fseqenlem1 9047 iundom2g 9564 eqresr 10160 cnref1o 12030 hashfun 13426 fsumcom2 14713 fprodcom2 14921 qredeu 15579 qnumdenbi 15659 crth 15690 prmreclem3 15829 imasaddfnlem 16396 dprd2da 18649 dprd2d2 18651 ucnima 22305 numclwwlk1lem2f1 27543 br8d 29760 xppreima2 29790 aciunf1lem 29802 ofpreima 29805 erdszelem9 31519 msubff1 31791 mvhf1 31794 brtp 31977 br8 31984 br6 31985 br4 31986 brsegle 32552 poimirlem4 33746 poimirlem9 33751 f1opr 33851 dib1dim 36975 diclspsn 37004 dihopelvalcpre 37058 dihmeetlem4preN 37116 dihmeetlem13N 37129 dih1dimatlem 37139 dihatlat 37144 pellexlem3 37921 pellex 37925 snhesn 38606 opelopab4 39292 |
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