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Theorem triun 4039
 Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
triun (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem triun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 3817 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.29 2569 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 (Tr 𝐵𝑦𝐵))
3 nfcv 2281 . . . . . . 7 𝑥𝑦
4 nfiu1 3843 . . . . . . 7 𝑥 𝑥𝐴 𝐵
53, 4nfss 3090 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
6 trss 4035 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 123 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
8 ssiun2 3856 . . . . . . . 8 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
9 sstr2 3104 . . . . . . . 8 (𝑦𝐵 → (𝐵 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
108, 9syl5com 29 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
117, 10syl5 32 . . . . . 6 (𝑥𝐴 → ((Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵))
125, 11rexlimi 2542 . . . . 5 (∃𝑥𝐴 (Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
132, 12syl 14 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
141, 13sylan2b 285 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1514ralrimiva 2505 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
16 dftr3 4030 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1715, 16sylibr 133 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071  ∪ ciun 3813  Tr wtr 4026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-iun 3815  df-tr 4027 This theorem is referenced by:  truni  4040
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