Theorem en1unielOLD 8794

Description: Obsolete version of en1uniel 8793 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 8713	. . . 4 ⊢ Rel ≈
2	1	brrelex1i 5643	. . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V)
3		uniexg 7585	. . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V)
4		snidg 4601	. . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆})
5	2, 3, 4	3syl 18	. 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆})
6		en1b 8788	. . 3 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
7	6	biimpi 215	. 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆})
8	5, 7	eleqtrrd 2844	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2110  Vcvv 3431  {csn 4567  ∪ cuni 4845   class class class wbr 5079  1_oc1o 8275   ≈ cen 8705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-1o 8282  df-en 8709

This theorem is referenced by: (None)