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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopth | Structured version Visualization version GIF version |
Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 5475), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.) |
Ref | Expression |
---|---|
altopth.1 | ⊢ 𝐴 ∈ V |
altopth.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
altopth | ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | altopth.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | altopth.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | altopthg 35243 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ⟪caltop 35232 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-altop 35234 |
This theorem is referenced by: altopthd 35248 altopelaltxp 35252 |
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