Theorem en1bOLD 9055

Description: Obsolete version of en1b 9054 as of 24-Sep-2024. (Contributed by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en1bOLD	⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Proof of Theorem en1bOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 9052	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4923	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 3477	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 4933	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	eqtrdi 2784	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 4644	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2771	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1925	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 216	. 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴})
11		id 22	. . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴})
12		snex 5437	. . . . . 6 ⊢ {∪ 𝐴} ∈ V
13	11, 12	eqeltrdi 2837	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V)
14	13	uniexd 7753	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
15		ensn1g 9050	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
16	14, 15	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o)
17	11, 16	eqbrtrd 5174	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o)
18	10, 17	impbii 208	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 205   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3473  {csn 4632  ∪ cuni 4912   class class class wbr 5152  1_oc1o 8486   ≈ cen 8967

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-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1o 8493  df-en 8971

This theorem is referenced by: (None)