ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prss GIF version

Theorem prss 3747
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 ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prss
StepHypRef Expression
1 unss 3309 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2 prss.1 . . . 4 𝐴 ∈ V
32snss 3726 . . 3 (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶)
4 prss.2 . . . 4 𝐵 ∈ V
54snss 3726 . . 3 (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶)
63, 5anbi12i 460 . 2 ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7 df-pr 3598 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
87sseq1i 3181 . 2 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
91, 6, 83bitr4i 212 1 ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2148  Vcvv 2737  cun 3127  wss 3129  {csn 3591  {cpr 3592
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598
This theorem is referenced by:  tpss  3756  prsspw  3763  exmidpw  6902  pw1ne1  7222
  Copyright terms: Public domain W3C validator