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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 5473 | . . 3 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
| 4 | id 22 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
| 5 | 3, 4 | eleqtrid 2847 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴} ∈ 〈𝐶, 𝐷〉) |
| 6 | 1 | sneqr 4840 | . . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶) |
| 7 | 6 | a1i 11 | . . 3 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶)) |
| 8 | oprcl 4899 | . . . . . . 7 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
| 9 | 8 | simpld 494 | . . . . . 6 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ V) |
| 10 | prid1g 4760 | . . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷}) | |
| 11 | 9, 10 | syl 17 | . . . . 5 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → 𝐶 ∈ {𝐶, 𝐷}) |
| 12 | eleq2 2830 | . . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷})) | |
| 13 | 11, 12 | syl5ibrcom 247 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴})) |
| 14 | elsni 4643 | . . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴) | |
| 15 | 14 | eqcomd 2743 | . . . 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 4871 | . . . . . 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 2843 | . . . 4 ⊢ ({𝐴} ∈ 〈𝐶, 𝐷〉 → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}}) |
| 21 | elpri 4649 | . . . 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 861 | . 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 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: opth 5481 dmsnopg 6233 funcnvsn 6616 oprabidw 7462 oprabid 7463 seqomlem2 8491 unxpdomlem3 9288 dfac5lem4 10166 dfac5lem4OLD 10168 dcomex 10487 canthwelem 10690 uzrdgfni 13999 fnpr2ob 17603 gsum2d2 19992 noseqrdgfn 28312 poimirlem9 37636 ichnreuop 47459 ichreuopeq 47460 |
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