Theorem sstr2 3249

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 3236	. . . 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 1928	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
4		ssalel 3229	. 2 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
5		ssalel 3229	. 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 1396   ∈ wcel 2205   ⊆ wss 3214

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  sstr  3250  sstri  3251  sseq1  3265  sseq2  3266  ssun3  3388  ssun4  3389  ssinss1  3454  ssdisj  3569  sspw  3687  triun  4226  trintssm  4229  sspwb  4337  exss  4348  relss  4842  funss  5376  funimass2  5439  fss  5526  fiintim  7204  sbthlem2  7241  sbthlemi3  7242  sbthlemi6  7245  lsslss  14641  lspss  14659  tgss  15040  tgcl  15041  tgss3  15055  clsss  15095  neiss  15127  ssnei2  15134  cnpnei  15196  cnptopco  15199  cnptoprest  15216  txcnp  15248  neibl  15468  metcnp3  15488  bj-nntrans  16833