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Theorem ssiun2sf 29710
Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
ssiun2sf.1 𝑥𝐴
ssiun2sf.2 𝑥𝐶
ssiun2sf.3 𝑥𝐷
ssiun2sf.4 (𝑥 = 𝐶𝐵 = 𝐷)
Assertion
Ref Expression
ssiun2sf (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Proof of Theorem ssiun2sf
StepHypRef Expression
1 ssiun2sf.2 . . 3 𝑥𝐶
2 ssiun2sf.1 . . . . 5 𝑥𝐴
31, 2nfel 2925 . . . 4 𝑥 𝐶𝐴
4 ssiun2sf.3 . . . . 5 𝑥𝐷
5 nfiu1 4682 . . . . 5 𝑥 𝑥𝐴 𝐵
64, 5nfss 3743 . . . 4 𝑥 𝐷 𝑥𝐴 𝐵
73, 6nfim 1976 . . 3 𝑥(𝐶𝐴𝐷 𝑥𝐴 𝐵)
8 eleq1 2837 . . . 4 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
9 ssiun2sf.4 . . . . 5 (𝑥 = 𝐶𝐵 = 𝐷)
109sseq1d 3779 . . . 4 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
118, 10imbi12d 333 . . 3 (𝑥 = 𝐶 → ((𝑥𝐴𝐵 𝑥𝐴 𝐵) ↔ (𝐶𝐴𝐷 𝑥𝐴 𝐵)))
12 ssiun2 4695 . . 3 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
131, 7, 11, 12vtoclgf 3413 . 2 (𝐶𝐴 → (𝐶𝐴𝐷 𝑥𝐴 𝐵))
1413pm2.43i 52 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wnfc 2899  wss 3721   ciun 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-iun 4654
This theorem is referenced by:  iundisj2f  29735  esum2dlem  30488  voliune  30626  volfiniune  30627
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