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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 33423 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 33422 | . 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 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ⟪caltop 33410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-sn 4560 df-pr 4562 df-altop 33412 |
This theorem is referenced by: altopthc 33425 |
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