Theorem en1OLD 8879

Description: Obsolete version of en1 8878 as of 23-Sep-2024. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en1OLD	⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem en1OLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 8366	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5097	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		bren 8806	. . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
4	2, 3	bitri 274	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
5		f1ocnv 6773	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
6		f1ofo 6768	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
7		forn 6736	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
9		f1of 6761	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
10		0ex 5248	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	fsn2 7058	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
12	11	simprbi 497	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
13	9, 12	syl 17	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
14	13	rneqd 5873	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
15	10	rnsnop 6156	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
16	14, 15	eqtrdi 2792	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
17	8, 16	eqtr3d 2778	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
18		fvex 6832	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
19		sneq 4582	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
20	19	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
21	18, 20	spcev 3554	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
22	5, 17, 21	3syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
23	22	exlimiv 1932	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
24	4, 23	sylbi 216	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
25		vex 3445	. . . . 5 ⊢ 𝑥 ∈ V
26	25	ensn1 8874	. . . 4 ⊢ {𝑥} ≈ 1o
27		breq1 5092	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
28	26, 27	mpbiri 257	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
29	28	exlimiv 1932	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
30	24, 29	impbii 208	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 205   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∅c0 4268  {csn 4572  ⟨cop 4578   class class class wbr 5089  ^◡ccnv 5613  ran crn 5615  ⟶wf 6469  –onto→wfo 6471  –1-1-onto→wf1o 6472  ‘cfv 6473  1_oc1o 8352   ≈ cen 8793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-1o 8359  df-en 8797

This theorem is referenced by: (None)