Theorem prss 3520

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 3117	. 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2		prss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snss 3494	. . 3 ⊢ (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶)
4		prss.2	. . . 4 ⊢ 𝐵 ∈ V
5	4	snss 3494	. . 3 ⊢ (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶)
6	3, 5	anbi12i 433	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7		df-pr 3382	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
8	7	sseq1i 2969	. 2 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
9	1, 6, 8	3bitr4i 201	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  {csn 3375  {cpr 3376

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382

This theorem is referenced by:  tpss  3529  prsspw  3536