Theorem ssiun2 4563

Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun2	⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem ssiun2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 3003	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2	1	ex 450	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
3		eliun 4524	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	2, 3	syl6ibr 242	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
5	4	ssrdv 3609	1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  ssiun2s  4564  disjxiun  4649  disjxiunOLD  4650  triun  4766  iunopeqop  4981  ixpf  7930  ixpiunwdom  8496  r1sdom  8637  r1val1  8649  rankuni2b  8716  rankval4  8730  cplem1  8752  domtriomlem  9264  ac6num  9301  iunfo  9361  iundom2g  9362  pwfseqlem3  9482  inar1  9597  tskuni  9605  iunconnlem  21230  ptclsg  21418  ovoliunlem1  23270  limciun  23658  ssiun2sf  29378  bnj906  31000  bnj999  31027  bnj1014  31030  bnj1408  31104  trpredrec  31738  iunmapss  39407  ssmapsn  39408  sge0iunmpt  40635  sge0iun  40636  voliunsge0lem  40689  omeiunltfirp  40733