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Mirrors > Home > MPE Home > Th. List > opthpr | Structured version Visualization version GIF version |
Description: An unordered pair has the ordered pair property (compare opth 5359) under certain conditions. (Contributed by NM, 27-Mar-2007.) |
Ref | Expression |
---|---|
preqr1.a | ⊢ 𝐴 ∈ V |
preqr1.b | ⊢ 𝐵 ∈ V |
preq12b.c | ⊢ 𝐶 ∈ V |
preq12b.d | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opthpr | ⊢ (𝐴 ≠ 𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preqr1.a | . . 3 ⊢ 𝐴 ∈ V | |
2 | preqr1.b | . . 3 ⊢ 𝐵 ∈ V | |
3 | preq12b.c | . . 3 ⊢ 𝐶 ∈ V | |
4 | preq12b.d | . . 3 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | preq12b 4773 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
6 | idd 24 | . . . 4 ⊢ (𝐴 ≠ 𝐷 → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
7 | df-ne 3014 | . . . . . 6 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷) | |
8 | pm2.21 123 | . . . . . 6 ⊢ (¬ 𝐴 = 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) | |
9 | 7, 8 | sylbi 218 | . . . . 5 ⊢ (𝐴 ≠ 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
10 | 9 | impd 411 | . . . 4 ⊢ (𝐴 ≠ 𝐷 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
11 | 6, 10 | jaod 853 | . . 3 ⊢ (𝐴 ≠ 𝐷 → (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
12 | orc 861 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) | |
13 | 11, 12 | impbid1 226 | . 2 ⊢ (𝐴 ≠ 𝐷 → (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
14 | 5, 13 | syl5bb 284 | 1 ⊢ (𝐴 ≠ 𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {cpr 4559 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-ne 3014 df-v 3494 df-un 3938 df-sn 4558 df-pr 4560 |
This theorem is referenced by: brdom7disj 9941 brdom6disj 9942 |
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