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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 5470 | . . 3 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
4 | id 22 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
5 | 3, 4 | eleqtrid 2831 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
6 | 1 | sneqr 4843 | . . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶) |
7 | 6 | a1i 11 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶)) |
8 | oprcl 4901 | . . . . . . 7 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
9 | 8 | simpld 493 | . . . . . 6 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
10 | prid1g 4766 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷}) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ {𝐶, 𝐷}) |
12 | eleq2 2814 | . . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷})) | |
13 | 11, 12 | syl5ibrcom 246 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴})) |
14 | elsni 4647 | . . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴) | |
15 | 14 | eqcomd 2731 | . . . 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 4873 | . . . . . 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 2827 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}}) |
21 | elpri 4653 | . . . 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 858 | . 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 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {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 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: opth 5478 dmsnopg 6219 funcnvsn 6604 oprabidw 7450 oprabid 7451 seqomlem2 8472 unxpdomlem3 9277 dfac5lem4 10151 dcomex 10472 canthwelem 10675 uzrdgfni 13959 fnpr2ob 17543 gsum2d2 19941 noseqrdgfn 28229 poimirlem9 37230 ichnreuop 46946 ichreuopeq 46947 |
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