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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 5325 | . . 3 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
4 | id 22 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
5 | 3, 4 | eleqtrid 2896 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
6 | 1 | sneqr 4731 | . . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶) |
7 | 6 | a1i 11 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶)) |
8 | oprcl 4791 | . . . . . . 7 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
9 | 8 | simpld 498 | . . . . . 6 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
10 | prid1g 4656 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷}) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ {𝐶, 𝐷}) |
12 | eleq2 2878 | . . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷})) | |
13 | 11, 12 | syl5ibrcom 250 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴})) |
14 | elsni 4542 | . . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴) | |
15 | 14 | eqcomd 2804 | . . . 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 4761 | . . . . . 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 2892 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}}) |
21 | elpri 4547 | . . . 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 857 | . 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 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 {cpr 4527 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: opth 5333 dmsnopg 6037 funcnvsn 6374 oprabidw 7166 oprabid 7167 seqomlem2 8070 unxpdomlem3 8708 dfac5lem4 9537 dcomex 9858 canthwelem 10061 uzrdgfni 13321 fnpr2ob 16823 gsum2d2 19087 poimirlem9 35066 ichnreuop 43989 ichreuopeq 43990 |
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