Theorem sdom1 9150

Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5310, ax-un 7680. (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 8404	. . . . . . 7 ⊢ 1o = {∅}
2	1	breq2i 5106	. . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅})
3		brdomi 8896	. . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅})
4		f1cdmsn 7228	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
5		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	ensn1 8958	. . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o
7		breq1 5101	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
8	6, 7	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
9	8	exlimiv 1931	. . . . . . . . . 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 1934	. . . . . . 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 8911	. . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
18	16, 17	sylnibr 329	. . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o)
19	18	necon4ai 2963	. 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅)
20		1n0 8415	. . . 4 ⊢ 1o ≠ ∅
21		1oex 8407	. . . . 5 ⊢ 1o ∈ V
22	21	0sdom 9036	. . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
23	20, 22	mpbir 231	. . 3 ⊢ ∅ ≺ 1o
24		breq1 5101	. . 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 1541  ∃wex 1780   ≠ wne 2932  ∅c0 4285  {csn 4580   class class class wbr 5098  –1-1→wf1 6489  1_oc1o 8390   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1o 8397  df-en 8884  df-dom 8885  df-sdom 8886

This theorem is referenced by:  modom  9151  frgpcyg  21528