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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopth1 | Structured version Visualization version GIF version |
Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.) |
Ref | Expression |
---|---|
altopth1 | ⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | altopthsn 35428 | . 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷})) | |
2 | sneqrg 4832 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶)) | |
3 | 2 | adantrd 491 | . 2 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐴 = 𝐶)) |
4 | 1, 3 | biimtrid 241 | 1 ⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {csn 4620 ⟪caltop 35423 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-sn 4621 df-pr 4623 df-altop 35425 |
This theorem is referenced by: (None) |
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