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Theorem prss 3511
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
prss.1 A V
prss.2 B V
Assertion
Ref Expression
prss ((A 𝐶 B 𝐶) ↔ {A, B} ⊆ 𝐶)

Proof of Theorem prss
StepHypRef Expression
1 unss 3111 . 2 (({A} ⊆ 𝐶 {B} ⊆ 𝐶) ↔ ({A} ∪ {B}) ⊆ 𝐶)
2 prss.1 . . . 4 A V
32snss 3485 . . 3 (A 𝐶 ↔ {A} ⊆ 𝐶)
4 prss.2 . . . 4 B V
54snss 3485 . . 3 (B 𝐶 ↔ {B} ⊆ 𝐶)
63, 5anbi12i 433 . 2 ((A 𝐶 B 𝐶) ↔ ({A} ⊆ 𝐶 {B} ⊆ 𝐶))
7 df-pr 3374 . . 3 {A, B} = ({A} ∪ {B})
87sseq1i 2963 . 2 ({A, B} ⊆ 𝐶 ↔ ({A} ∪ {B}) ⊆ 𝐶)
91, 6, 83bitr4i 201 1 ((A 𝐶 B 𝐶) ↔ {A, B} ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1390  Vcvv 2551  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:  tpss  3520  prsspw  3527
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