Theorem en1unielOLD 9043

Description: Obsolete version of en1uniel 9042 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 8958	. . . 4 ⊢ Rel ≈
2	1	brrelex1i 5728	. . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V)
3		uniexg 7737	. . 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 9037	. . 3 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
7	6	biimpi 215	. 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆})
8	5, 7	eleqtrrd 2831	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {csn 4624  ∪ cuni 4903   class class class wbr 5142  1_oc1o 8471   ≈ cen 8950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  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 6369  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 8478  df-en 8954

This theorem is referenced by: (None)