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Theorem ssiun 3690
 Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun (x A 𝐶B𝐶 x A B)
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem ssiun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . 5 (𝐶B → (y 𝐶y B))
21reximi 2410 . . . 4 (x A 𝐶Bx A (y 𝐶y B))
3 r19.37av 2457 . . . 4 (x A (y 𝐶y B) → (y 𝐶x A y B))
42, 3syl 14 . . 3 (x A 𝐶B → (y 𝐶x A y B))
5 eliun 3652 . . 3 (y x A Bx A y B)
64, 5syl6ibr 151 . 2 (x A 𝐶B → (y 𝐶y x A B))
76ssrdv 2945 1 (x A 𝐶B𝐶 x A B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ∪ ciun 3648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650 This theorem is referenced by:  iunss2  3693  iunpwss  3734  iunpw  4177
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