Theorem en1unielOLD 9050

Description: Obsolete version of en1uniel 9049 as of 24-Sep-2024. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en1unielOLD	⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Proof of Theorem en1unielOLD

Step	Hyp	Ref	Expression
1		relen 8965	. . . 4 ⊢ Rel ≈
2	1	brrelex1i 5728	. . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V)
3		uniexg 7742	. . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V)
4		snidg 4658	. . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆})
5	2, 3, 4	3syl 18	. 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆})
6		en1b 9044	. . 3 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
7	6	biimpi 215	. 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆})
8	5, 7	eleqtrrd 2828	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3463  {csn 4624  ∪ cuni 4903   class class class wbr 5143  1_oc1o 8476   ≈ cen 8957

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8483  df-en 8961

This theorem is referenced by: (None)