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Theorem iunidOLD 5057
Description: Obsolete version of iunid 5056 as of 15-Jan-2025. (Contributed by NM, 6-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
iunidOLD 𝑥𝐴 {𝑥} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem iunidOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sn 4624 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 2013 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2796 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2754 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 11 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 5011 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 5047 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 3224 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2796 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2865 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2760 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2754 1 𝑥𝐴 {𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2703  wrex 3064  {csn 4623   ciun 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-sn 4624  df-iun 4992
This theorem is referenced by: (None)
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