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Theorem prssg 3512
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
prssg ((A 𝑉 B 𝑊) → ((A 𝐶 B 𝐶) ↔ {A, B} ⊆ 𝐶))

Proof of Theorem prssg
StepHypRef Expression
1 snssg 3491 . . 3 (A 𝑉 → (A 𝐶 ↔ {A} ⊆ 𝐶))
2 snssg 3491 . . 3 (B 𝑊 → (B 𝐶 ↔ {B} ⊆ 𝐶))
31, 2bi2anan9 538 . 2 ((A 𝑉 B 𝑊) → ((A 𝐶 B 𝐶) ↔ ({A} ⊆ 𝐶 {B} ⊆ 𝐶)))
4 unss 3111 . . 3 (({A} ⊆ 𝐶 {B} ⊆ 𝐶) ↔ ({A} ∪ {B}) ⊆ 𝐶)
5 df-pr 3374 . . . 4 {A, B} = ({A} ∪ {B})
65sseq1i 2963 . . 3 ({A, B} ⊆ 𝐶 ↔ ({A} ∪ {B}) ⊆ 𝐶)
74, 6bitr4i 176 . 2 (({A} ⊆ 𝐶 {B} ⊆ 𝐶) ↔ {A, B} ⊆ 𝐶)
83, 7syl6bb 185 1 ((A 𝑉 B 𝑊) → ((A 𝐶 B 𝐶) ↔ {A, B} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  cun 2909  wss 2911  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374
This theorem is referenced by:  prssi  3513  prsspwg  3514
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