![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfopg | Structured version Visualization version GIF version |
Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
dfopg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | elex 3493 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
3 | dfopif 4871 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
4 | iftrue 4535 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}}) | |
5 | 3, 4 | eqtrid 2785 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) |
6 | 1, 2, 5 | syl2an 597 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 ifcif 4529 {csn 4629 {cpr 4631 ⟨cop 4635 |
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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-op 4636 |
This theorem is referenced by: dfop 4873 opidg 4893 elopg 5467 opnz 5474 opth1 5476 opth 5477 0nelop 5497 opeqsng 5504 opwf 9807 rankopb 9847 wunop 10717 tskop 10766 gruop 10800 |
Copyright terms: Public domain | W3C validator |