Theorem ssnel 4413

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 4385	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		ssel 3033	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
3	1, 2	mtoi 628	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1445   ⊆ wss 3013

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-sn 3472

This theorem is referenced by:  nntri1  6297