Theorem onnmin 4311

Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)

Assertion

Ref	Expression
onnmin	⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)

Proof of Theorem onnmin

Step	Hyp	Ref	Expression
1		intss1 3651	. . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵)
2		elirr 4284	. . . 4 ⊢ ¬ 𝐵 ∈ 𝐵
3		ssel 2993	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝐵 → (𝐵 ∈ ∩ 𝐴 → 𝐵 ∈ 𝐵))
4	2, 3	mtoi 622	. . 3 ⊢ (∩ 𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ ∩ 𝐴)
5	1, 4	syl 14	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ ∩ 𝐴)
6	5	adantl 271	1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∈ wcel 1433   ⊆ wss 2973  ∩ cint 3636  Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-sn 3404  df-int 3637

This theorem is referenced by: (None)