Theorem sstr2 3202

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 3189	. . . 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 1903	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
4		ssalel 3183	. 2 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
5		ssalel 3183	. 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 1371   ∈ wcel 2177   ⊆ wss 3168

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3174  df-ss 3181

This theorem is referenced by:  sstr  3203  sstri  3204  sseq1  3218  sseq2  3219  ssun3  3340  ssun4  3341  ssinss1  3404  ssdisj  3519  triun  4160  trintssm  4163  sspwb  4265  exss  4276  relss  4767  funss  5296  funimass2  5358  fss  5444  fiintim  7040  sbthlem2  7072  sbthlemi3  7073  sbthlemi6  7076  lsslss  14193  lspss  14211  tgss  14585  tgcl  14586  tgss3  14600  clsss  14640  neiss  14672  ssnei2  14679  cnpnei  14741  cnptopco  14744  cnptoprest  14761  txcnp  14793  neibl  15013  metcnp3  15033  bj-nntrans  16001