Theorem intmin2 3872

Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
intmin2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intmin2	⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem intmin2

Step	Hyp	Ref	Expression
1		rabab 2760	. . 3 ⊢ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = {𝑥 ∣ 𝐴 ⊆ 𝑥}
2	1	inteqi 3850	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥}
3		intmin2.1	. . 3 ⊢ 𝐴 ∈ V
4		intmin 3866	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴)
5	3, 4	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥} = 𝐴
6	2, 5	eqtr3i 2200	1 ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2148  {cab 2163  {crab 2459  Vcvv 2739   ⊆ wss 3131  ∩ cint 3846

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-ral 2460  df-rab 2464  df-v 2741  df-in 3137  df-ss 3144  df-int 3847

This theorem is referenced by: (None)