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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 35598 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 35597 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⟪caltop 35585 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-sn 4633 df-pr 4635 df-altop 35587 |
This theorem is referenced by: altopthc 35600 |
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