Theorem prss 4756

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.) (Proof shortened by JJ, 23-Jul-2021.)

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		prss.1	. 2 ⊢ 𝐴 ∈ V
2		prss.2	. 2 ⊢ 𝐵 ∈ V
3		prssg 4755	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
4	1, 2, 3	mp2an 690	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  Vcvv 3497   ⊆ wss 3939  {cpr 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-in 3946  df-ss 3955  df-sn 4571  df-pr 4573

This theorem is referenced by:  tpss  4771  uniintsn  4916  pwssun  5459  xpsspw  5685  dffv2  6759  fiint  8798  wunex2  10163  hashfun  13801  fun2dmnop0  13855  prdsle  16738  prdsless  16739  prdsleval  16753  pwsle  16768  acsfn2  16937  joinfval  17614  joindmss  17620  meetfval  17628  meetdmss  17634  clatl  17729  ipoval  17767  ipolerval  17769  eqgfval  18331  eqgval  18332  gaorb  18440  pmtrrn2  18591  efgcpbllema  18883  frgpuplem  18901  isnzr2hash  20040  ltbval  20255  ltbwe  20256  opsrle  20259  opsrtoslem1  20267  thlle  20844  isphtpc  23601  axlowdimlem4  26734  structgrssvtx  26812  structgrssiedg  26813  umgredg  26926  wlk1walk  27423  wlkonl1iedg  27450  wlkdlem2  27468  3wlkdlem6  27947  frcond2  28049  frcond3  28051  nfrgr2v  28054  frgr3vlem1  28055  frgr3vlem2  28056  2pthfrgrrn  28064  frgrncvvdeqlem2  28082  shincli  29142  chincli  29240  lsmsnorb  30949  coinfliprv  31744  altxpsspw  33442  mnurndlem1  40623  fourierdlem103  42501  fourierdlem104  42502  nnsum3primes4  43960