Theorem ssnel 4630

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 4602	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		ssel 3191	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
3	1, 2	mtoi 666	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2177   ⊆ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-setind 4598

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-sn 3644

This theorem is referenced by:  nntri1  6600  pw1ne3  7371  3nelsucpw1  7375  3nsssucpw1  7377  nninfctlemfo  12446