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Theorem iunxdif3 4981
 Description: An indexed union where some terms are the empty set. See iunxdif2 4941. (Contributed by Thierry Arnoux, 4-May-2020.)
Hypothesis
Ref Expression
iunxdif3.1 𝑥𝐸
Assertion
Ref Expression
iunxdif3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem iunxdif3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 inss2 4156 . . . . . 6 (𝐴𝐸) ⊆ 𝐸
2 nfcv 2955 . . . . . . . . . 10 𝑥𝐴
3 iunxdif3.1 . . . . . . . . . 10 𝑥𝐸
42, 3nfin 4143 . . . . . . . . 9 𝑥(𝐴𝐸)
54, 3ssrexf 3979 . . . . . . . 8 ((𝐴𝐸) ⊆ 𝐸 → (∃𝑥 ∈ (𝐴𝐸)𝑦𝐵 → ∃𝑥𝐸 𝑦𝐵))
6 eliun 4886 . . . . . . . 8 (𝑦 𝑥 ∈ (𝐴𝐸)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐸)𝑦𝐵)
7 eliun 4886 . . . . . . . 8 (𝑦 𝑥𝐸 𝐵 ↔ ∃𝑥𝐸 𝑦𝐵)
85, 6, 73imtr4g 299 . . . . . . 7 ((𝐴𝐸) ⊆ 𝐸 → (𝑦 𝑥 ∈ (𝐴𝐸)𝐵𝑦 𝑥𝐸 𝐵))
98ssrdv 3921 . . . . . 6 ((𝐴𝐸) ⊆ 𝐸 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵)
101, 9ax-mp 5 . . . . 5 𝑥 ∈ (𝐴𝐸)𝐵 𝑥𝐸 𝐵
11 iuneq2 4901 . . . . . 6 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = 𝑥𝐸 ∅)
12 iun0 4949 . . . . . 6 𝑥𝐸 ∅ = ∅
1311, 12eqtrdi 2849 . . . . 5 (∀𝑥𝐸 𝐵 = ∅ → 𝑥𝐸 𝐵 = ∅)
1410, 13sseqtrid 3967 . . . 4 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅)
15 ss0 4306 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ⊆ ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1614, 15syl 17 . . 3 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = ∅)
1716uneq1d 4089 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵))
18 iunxun 4980 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵)
19 inundif 4385 . . . . 5 ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
2019nfth 1803 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴
212, 3nfdif 4053 . . . . . . 7 𝑥(𝐴𝐸)
224, 21nfun 4092 . . . . . 6 𝑥((𝐴𝐸) ∪ (𝐴𝐸))
23 id 22 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 → ((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴)
24 eqidd 2799 . . . . . 6 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴𝐵 = 𝐵)
2520, 22, 2, 23, 24iuneq12df 4908 . . . . 5 (((𝐴𝐸) ∪ (𝐴𝐸)) = 𝐴 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵)
2619, 25ax-mp 5 . . . 4 𝑥 ∈ ((𝐴𝐸) ∪ (𝐴𝐸))𝐵 = 𝑥𝐴 𝐵
2718, 26eqtr3i 2823 . . 3 ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵
2827a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → ( 𝑥 ∈ (𝐴𝐸)𝐵 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥𝐴 𝐵)
29 uncom 4080 . . . 4 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅)
30 un0 4298 . . . 4 ( 𝑥 ∈ (𝐴𝐸)𝐵 ∪ ∅) = 𝑥 ∈ (𝐴𝐸)𝐵
3129, 30eqtri 2821 . . 3 (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵
3231a1i 11 . 2 (∀𝑥𝐸 𝐵 = ∅ → (∅ ∪ 𝑥 ∈ (𝐴𝐸)𝐵) = 𝑥 ∈ (𝐴𝐸)𝐵)
3317, 28, 323eqtr3rd 2842 1 (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ∪ ciun 4882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-iun 4884 This theorem is referenced by:  aciunf1  30436  suppovss  30453  ovnsubadd2lem  43327
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