Theorem prss 3623

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

Step	Hyp	Ref	Expression
1		unss 3197	. 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2		prss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snss 3596	. . 3 ⊢ (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶)
4		prss.2	. . . 4 ⊢ 𝐵 ∈ V
5	4	snss 3596	. . 3 ⊢ (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶)
6	3, 5	anbi12i 451	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7		df-pr 3481	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
8	7	sseq1i 3073	. 2 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
9	1, 6, 8	3bitr4i 211	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1448  Vcvv 2641   ∪ cun 3019   ⊆ wss 3021  {csn 3474  {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481

This theorem is referenced by:  tpss  3632  prsspw  3639  exmidpw  6731