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Mirrors > Home > NFE Home > Th. List > opkabssvvki | GIF version |
Description: Any Kuratowski ordered pair abstraction is a subset of (V ×k V). (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
opkabssvvki.1 | ⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)} |
Ref | Expression |
---|---|
opkabssvvki | ⊢ A ⊆ (V ×k V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opkabssvvki.1 | . 2 ⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)} | |
2 | opkabssvvk 4208 | . 2 ⊢ {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)} ⊆ (V ×k V) | |
3 | 1, 2 | eqsstri 3301 | 1 ⊢ A ⊆ (V ×k V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 {cab 2339 Vcvv 2859 ⊆ wss 3257 ⟪copk 4057 ×k cxpk 4174 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 |
This theorem is referenced by: xpkssvvk 4210 sikssvvk 4266 cnvkssvvk 4275 ssetkssvvk 4278 ins2kss 4279 ins3kss 4280 idkssvvk 4281 |
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