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Theorem iunxsng 4754
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsng (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4676 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 iunxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2825 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43rexsng 4363 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
51, 4syl5bb 272 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
65eqrdv 2758 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wrex 3051  {csn 4321   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-sn 4322  df-iun 4674
This theorem is referenced by:  iunxsn  4755  iunxprg  4759  disjiun2  39725  carageniuncllem1  41241  caratheodorylem1  41246
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