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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopthb | Structured version Visualization version GIF version |
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34600 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
Ref | Expression |
---|---|
altopthb.1 | ⊢ 𝐴 ∈ V |
altopthb.2 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
altopthb | ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | altopthb.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | altopthb.2 | . 2 ⊢ 𝐷 ∈ V | |
3 | altopthbg 34599 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⟪caltop 34587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-sn 4588 df-pr 4590 df-altop 34589 |
This theorem is referenced by: altopthc 34602 |
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