Theorem sstr2 3232

Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
sstr2	⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))

Proof of Theorem sstr2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3219	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	imim1d 75	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
3	2	alimdv 1925	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
4		ssalel 3213	. 2 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
5		ssalel 3213	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
6	3, 4, 5	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))

Syntax hints:   → wi 4  ∀wal 1393   ∈ wcel 2200   ⊆ wss 3198

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  sstr  3233  sstri  3234  sseq1  3248  sseq2  3249  ssun3  3370  ssun4  3371  ssinss1  3434  ssdisj  3549  triun  4198  trintssm  4201  sspwb  4306  exss  4317  relss  4811  funss  5343  funimass2  5405  fss  5491  fiintim  7118  sbthlem2  7151  sbthlemi3  7152  sbthlemi6  7155  lsslss  14388  lspss  14406  tgss  14780  tgcl  14781  tgss3  14795  clsss  14835  neiss  14867  ssnei2  14874  cnpnei  14936  cnptopco  14939  cnptoprest  14956  txcnp  14988  neibl  15208  metcnp3  15228  bj-nntrans  16496