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Mirrors > Home > MPE Home > Th. List > opth1 | Structured version Visualization version GIF version |
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 | ⊢ 𝐴 ∈ V |
opth1.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opth1 | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | opth1.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opi1 5351 | . . 3 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
4 | id 22 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
5 | 3, 4 | eleqtrid 2916 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
6 | 1 | sneqr 4763 | . . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶) |
7 | 6 | a1i 11 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶)) |
8 | oprcl 4821 | . . . . . . 7 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
9 | 8 | simpld 495 | . . . . . 6 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
10 | prid1g 4688 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷}) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ {𝐶, 𝐷}) |
12 | eleq2 2898 | . . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷})) | |
13 | 11, 12 | syl5ibrcom 248 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴})) |
14 | elsni 4574 | . . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴) | |
15 | 14 | eqcomd 2824 | . . . 4 ⊢ (𝐶 ∈ {𝐴} → 𝐴 = 𝐶) |
16 | 13, 15 | syl6 35 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶)) |
17 | id 22 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) | |
18 | dfopg 4793 | . . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐷}}) | |
19 | 8, 18 | syl 17 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 〈𝐶, 𝐷〉 = {{𝐶}, {𝐶, 𝐷}}) |
20 | 17, 19 | eleqtrd 2912 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}}) |
21 | elpri 4579 | . . . 4 ⊢ ({𝐴} ∈ {{𝐶}, {𝐶, 𝐷}} → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷})) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷})) |
23 | 7, 16, 22 | mpjaod 854 | . 2 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) |
24 | 5, 23 | syl 17 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 {cpr 4559 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 |
This theorem is referenced by: opth 5359 dmsnopg 6063 funcnvsn 6397 oprabidw 7176 oprabid 7177 seqomlem2 8076 unxpdomlem3 8712 dfac5lem4 9540 dcomex 9857 canthwelem 10060 uzrdgfni 13314 fnpr2ob 16819 gsum2d2 19023 poimirlem9 34782 ichnreuop 43511 ichreuopeq 43512 |
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