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Theorem relop 5305
Description: A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
Hypotheses
Ref Expression
relop.1 𝐴 ∈ V
relop.2 𝐵 ∈ V
Assertion
Ref Expression
relop (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relop
Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 5150 . 2 (Rel ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ ⊆ (V × V))
2 dfss2 3624 . . . . 5 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
3 relop.1 . . . . . . . . . 10 𝐴 ∈ V
4 relop.2 . . . . . . . . . 10 𝐵 ∈ V
53, 4elop 4965 . . . . . . . . 9 (𝑧 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}))
6 elvv 5211 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
75, 6imbi12i 339 . . . . . . . 8 ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
8 jaob 839 . . . . . . . 8 (((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
97, 8bitri 264 . . . . . . 7 ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
109albii 1787 . . . . . 6 (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ∀𝑧((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
11 19.26 1838 . . . . . 6 (∀𝑧((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
1210, 11bitri 264 . . . . 5 (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
132, 12bitri 264 . . . 4 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
14 snex 4938 . . . . . . 7 {𝐴} ∈ V
15 eqeq1 2655 . . . . . . . 8 (𝑧 = {𝐴} → (𝑧 = {𝐴} ↔ {𝐴} = {𝐴}))
16 eqeq1 2655 . . . . . . . . . 10 (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴} = ⟨𝑥, 𝑦⟩))
17 eqcom 2658 . . . . . . . . . . 11 ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴})
18 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
19 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
2018, 19, 3opeqsn 4996 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {𝐴} ↔ (𝑥 = 𝑦𝐴 = {𝑥}))
2117, 20bitri 264 . . . . . . . . . 10 ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦𝐴 = {𝑥}))
2216, 21syl6bb 276 . . . . . . . . 9 (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦𝐴 = {𝑥})))
23222exbidv 1892 . . . . . . . 8 (𝑧 = {𝐴} → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
2415, 23imbi12d 333 . . . . . . 7 (𝑧 = {𝐴} → ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}))))
2514, 24spcv 3330 . . . . . 6 (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
26 sneq 4220 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤} = {𝑥})
2726eqeq2d 2661 . . . . . . . 8 (𝑤 = 𝑥 → (𝐴 = {𝑤} ↔ 𝐴 = {𝑥}))
2827cbvexv 2311 . . . . . . 7 (∃𝑤 𝐴 = {𝑤} ↔ ∃𝑥 𝐴 = {𝑥})
29 ax6evr 1988 . . . . . . . . 9 𝑦 𝑥 = 𝑦
30 19.41v 1917 . . . . . . . . 9 (∃𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ (∃𝑦 𝑥 = 𝑦𝐴 = {𝑥}))
3129, 30mpbiran 973 . . . . . . . 8 (∃𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ 𝐴 = {𝑥})
3231exbii 1814 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ ∃𝑥 𝐴 = {𝑥})
33 eqid 2651 . . . . . . . 8 {𝐴} = {𝐴}
3433a1bi 351 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
3528, 32, 343bitr2ri 289 . . . . . 6 (({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})) ↔ ∃𝑤 𝐴 = {𝑤})
3625, 35sylib 208 . . . . 5 (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑤 𝐴 = {𝑤})
37 eqid 2651 . . . . . 6 {𝐴, 𝐵} = {𝐴, 𝐵}
38 prex 4939 . . . . . . 7 {𝐴, 𝐵} ∈ V
39 eqeq1 2655 . . . . . . . 8 (𝑧 = {𝐴, 𝐵} → (𝑧 = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴, 𝐵}))
40 eqeq1 2655 . . . . . . . . 9 (𝑧 = {𝐴, 𝐵} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
41402exbidv 1892 . . . . . . . 8 (𝑧 = {𝐴, 𝐵} → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
4239, 41imbi12d 333 . . . . . . 7 (𝑧 = {𝐴, 𝐵} → ((𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)))
4338, 42spcv 3330 . . . . . 6 (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
4437, 43mpi 20 . . . . 5 (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)
45 eqcom 2658 . . . . . . . . . 10 ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴, 𝐵})
4618, 19, 3, 4opeqpr 4997 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = {𝐴, 𝐵} ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
4745, 46bitri 264 . . . . . . . . 9 ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
48 idd 24 . . . . . . . . . 10 (𝐴 = {𝑤} → ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
49 eqtr2 2671 . . . . . . . . . . . . . 14 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → {𝑥, 𝑦} = {𝑤})
50 vex 3234 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
5118, 19, 50preqsn 4424 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} = {𝑤} ↔ (𝑥 = 𝑦𝑦 = 𝑤))
5251simplbi 475 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} = {𝑤} → 𝑥 = 𝑦)
5349, 52syl 17 . . . . . . . . . . . . 13 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → 𝑥 = 𝑦)
54 dfsn2 4223 . . . . . . . . . . . . . . . . . . . 20 {𝑥} = {𝑥, 𝑥}
55 preq2 4301 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
5654, 55syl5req 2698 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
5756eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {𝑥}))
5854, 55syl5eq 2697 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → {𝑥} = {𝑥, 𝑦})
5958eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑥, 𝑦}))
6057, 59anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) ↔ (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6160biimpd 219 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6261expd 451 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6362com12 32 . . . . . . . . . . . . . 14 (𝐴 = {𝑥, 𝑦} → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6463adantr 480 . . . . . . . . . . . . 13 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6553, 64mpd 15 . . . . . . . . . . . 12 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6665expcom 450 . . . . . . . . . . 11 (𝐴 = {𝑤} → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6766impd 446 . . . . . . . . . 10 (𝐴 = {𝑤} → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6848, 67jaod 394 . . . . . . . . 9 (𝐴 = {𝑤} → (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6947, 68syl5bi 232 . . . . . . . 8 (𝐴 = {𝑤} → ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
70692eximdv 1888 . . . . . . 7 (𝐴 = {𝑤} → (∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
7170exlimiv 1898 . . . . . 6 (∃𝑤 𝐴 = {𝑤} → (∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
7271imp 444 . . . . 5 ((∃𝑤 𝐴 = {𝑤} ∧ ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
7336, 44, 72syl2an 493 . . . 4 ((∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
7413, 73sylbi 207 . . 3 (⟨𝐴, 𝐵⟩ ⊆ (V × V) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
75 simpr 476 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = {𝐴})
76 equid 1985 . . . . . . . . . . . . . 14 𝑥 = 𝑥
7776jctl 563 . . . . . . . . . . . . 13 (𝐴 = {𝑥} → (𝑥 = 𝑥𝐴 = {𝑥}))
7818, 18, 3opeqsn 4996 . . . . . . . . . . . . 13 (⟨𝑥, 𝑥⟩ = {𝐴} ↔ (𝑥 = 𝑥𝐴 = {𝑥}))
7977, 78sylibr 224 . . . . . . . . . . . 12 (𝐴 = {𝑥} → ⟨𝑥, 𝑥⟩ = {𝐴})
8079adantr 480 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ⟨𝑥, 𝑥⟩ = {𝐴})
8175, 80eqtr4d 2688 . . . . . . . . . 10 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = ⟨𝑥, 𝑥⟩)
82 opeq12 4435 . . . . . . . . . . . 12 ((𝑤 = 𝑥𝑣 = 𝑥) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑥⟩)
8382eqeq2d 2661 . . . . . . . . . . 11 ((𝑤 = 𝑥𝑣 = 𝑥) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
8418, 18, 83spc2ev 3332 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑥⟩ → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
8581, 84syl 17 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
8685adantlr 751 . . . . . . . 8 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
87 preq12 4302 . . . . . . . . . . . 12 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → {𝐴, 𝐵} = {{𝑥}, {𝑥, 𝑦}})
8887eqeq2d 2661 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 = {𝐴, 𝐵} ↔ 𝑧 = {{𝑥}, {𝑥, 𝑦}}))
8988biimpa 500 . . . . . . . . . 10 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = {{𝑥}, {𝑥, 𝑦}})
9018, 19dfop 4432 . . . . . . . . . 10 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
9189, 90syl6eqr 2703 . . . . . . . . 9 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = ⟨𝑥, 𝑦⟩)
92 opeq12 4435 . . . . . . . . . . 11 ((𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
9392eqeq2d 2661 . . . . . . . . . 10 ((𝑤 = 𝑥𝑣 = 𝑦) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
9418, 19, 93spc2ev 3332 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9591, 94syl 17 . . . . . . . 8 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9686, 95jaodan 843 . . . . . . 7 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵})) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9796ex 449 . . . . . 6 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩))
98 elvv 5211 . . . . . 6 (𝑧 ∈ (V × V) ↔ ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9997, 5, 983imtr4g 285 . . . . 5 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
10099ssrdv 3642 . . . 4 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
101100exlimivv 1900 . . 3 (∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
10274, 101impbii 199 . 2 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
1031, 102bitri 264 1 (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  wss 3607  {csn 4210  {cpr 4212  cop 4216   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  funopg  5960
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