Theorem sdom1 9166

Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5315, ax-un 7691. (Revised by BTernaryTau, 12-Dec-2024.)

Assertion

Ref	Expression
sdom1	⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅)

Proof of Theorem sdom1

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 8418	. . . . . . 7 ⊢ 1o = {∅}
2	1	breq2i 5110	. . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅})
3		brdomi 8908	. . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅})
4		f1cdmsn 7239	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
5		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	ensn1 8969	. . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o
7		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
8	6, 7	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
9	8	exlimiv 1930	. . . . . . . . . 10 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
10	4, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 1o)
11	10	expcom 413	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o))
12	11	exlimdv 1933	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o))
13	3, 12	syl5 34	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ {∅} → 𝐴 ≈ 1o))
14	2, 13	biimtrid 242	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ 1o → 𝐴 ≈ 1o))
15		iman 401	. . . . 5 ⊢ ((𝐴 ≼ 1o → 𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
16	14, 15	sylib 218	. . . 4 ⊢ (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
17		brsdom 8923	. . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
18	16, 17	sylnibr 329	. . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o)
19	18	necon4ai 2956	. 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅)
20		1n0 8429	. . . 4 ⊢ 1o ≠ ∅
21		1oex 8421	. . . . 5 ⊢ 1o ∈ V
22	21	0sdom 9049	. . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
23	20, 22	mpbir 231	. . 3 ⊢ ∅ ≺ 1o
24		breq1 5105	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o))
25	23, 24	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o)
26	19, 25	impbii 209	1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ≠ wne 2925  ∅c0 4292  {csn 4585   class class class wbr 5102  –1-1→wf1 6496  1_oc1o 8404   ≈ cen 8892   ≼ cdom 8893   ≺ csdm 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-1o 8411  df-en 8896  df-dom 8897  df-sdom 8898

This theorem is referenced by:  modom  9167  frgpcyg  21515