Theorem onnmin 4604

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 3889	. . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵)
2		elirr 4577	. . . 4 ⊢ ¬ 𝐵 ∈ 𝐵
3		ssel 3177	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝐵 → (𝐵 ∈ ∩ 𝐴 → 𝐵 ∈ 𝐵))
4	2, 3	mtoi 665	. . 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 2167   ⊆ wss 3157  ∩ cint 3874  Oncon0 4398

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-sn 3628  df-int 3875

This theorem is referenced by: (None)