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Theorem triun 4098
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 3875 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.29 2607 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 (Tr 𝐵𝑦𝐵))
3 nfcv 2312 . . . . . . 7 𝑥𝑦
4 nfiu1 3901 . . . . . . 7 𝑥 𝑥𝐴 𝐵
53, 4nfss 3140 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
6 trss 4094 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 123 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
8 ssiun2 3914 . . . . . . . 8 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
9 sstr2 3154 . . . . . . . 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 2580 . . . . 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 2543 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
16 dftr3 4089 . 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 2141  wral 2448  wrex 2449  wss 3121   ciun 3871  Tr wtr 4085
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-iun 3873  df-tr 4086
This theorem is referenced by:  truni  4099
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