Theorem prss 3866

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 3403	. 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2		prss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snss 3845	. . 3 ⊢ (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶)
4		prss.2	. . . 4 ⊢ 𝐵 ∈ V
5	4	snss 3845	. . 3 ⊢ (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶)
6	3, 5	anbi12i 464	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7		df-pr 3712	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
8	7	sseq1i 3274	. 2 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
9	1, 6, 8	3bitr4i 212	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3705  {cpr 3706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712

This theorem is referenced by:  tpss  3878  prsspw  3885  exmidpw  7205  pw1ne1  7578  releqgg  14000  eqgex  14001  eqgfval  14002  eqgval  14003  prdsex  14149  prdsval  14150  prdsbaslemss  14151  umgredg  16300