Theorem ssnel 4546

Description: Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)

Assertion

Ref	Expression
ssnel	⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Proof of Theorem ssnel

Step	Hyp	Ref	Expression
1		elirr 4518	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		ssel 3136	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
3	1, 2	mtoi 654	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2136   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-sn 3582

This theorem is referenced by:  nntri1  6464  pw1ne3  7186  3nelsucpw1  7190  3nsssucpw1  7192