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

Step	Hyp	Ref	Expression
1		snssg 3491	. . 3 ⊢ (A ∈ 𝑉 → (A ∈ 𝐶 ↔ {A} ⊆ 𝐶))
2		snssg 3491	. . 3 ⊢ (B ∈ 𝑊 → (B ∈ 𝐶 ↔ {B} ⊆ 𝐶))
3	1, 2	bi2anan9 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})
6	5	sseq1i 2963	. . 3 ⊢ ({A, B} ⊆ 𝐶 ↔ ({A} ∪ {B}) ⊆ 𝐶)
7	4, 6	bitr4i 176	. 2 ⊢ (({A} ⊆ 𝐶 ∧ {B} ⊆ 𝐶) ↔ {A, B} ⊆ 𝐶)
8	3, 7	syl6bb 185	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((A ∈ 𝐶 ∧ B ∈ 𝐶) ↔ {A, B} ⊆ 𝐶))

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-bndl 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