Theorem onnmin 4637

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 3917	. . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵)
2		elirr 4610	. . . 4 ⊢ ¬ 𝐵 ∈ 𝐵
3		ssel 3198	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝐵 → (𝐵 ∈ ∩ 𝐴 → 𝐵 ∈ 𝐵))
4	2, 3	mtoi 668	. . 3 ⊢ (∩ 𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ ∩ 𝐴)
5	1, 4	syl 14	. 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ ∩ 𝐴)
6	5	adantl 277	1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2180   ⊆ wss 3177  ∩ cint 3902  Oncon0 4431

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-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-v 2781  df-dif 3179  df-in 3183  df-ss 3190  df-sn 3652  df-int 3903

This theorem is referenced by: (None)