Theorem sdom1 9151

Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5295, ax-un 7679. (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 8403	. . . . . . 7 ⊢ 1o = {∅}
2	1	breq2i 5081	. . . . . 6 ⊢ (𝐴 ≼ 1o ↔ 𝐴 ≼ {∅})
3		brdomi 8897	. . . . . . 7 ⊢ (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴–1-1→{∅})
4		f1cdmsn 7227	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
5		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	ensn1 8959	. . . . . . . . . . . 12 ⊢ {𝑥} ≈ 1o
7		breq1 5076	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
8	6, 7	mpbiri 259	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
9	8	exlimiv 1937	. . . . . . . . . 10 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
10	4, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 1o)
11	10	expcom 414	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o))
12	11	exlimdv 1940	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:𝐴–1-1→{∅} → 𝐴 ≈ 1o))
13	3, 12	syl5 34	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ {∅} → 𝐴 ≈ 1o))
14	2, 13	biimtrid 243	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ≼ 1o → 𝐴 ≈ 1o))
15		iman 402	. . . . 5 ⊢ ((𝐴 ≼ 1o → 𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
16	14, 15	sylib 219	. . . 4 ⊢ (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
17		brsdom 8912	. . . 4 ⊢ (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
18	16, 17	sylnibr 330	. . 3 ⊢ (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o)
19	18	necon4ai 2965	. 2 ⊢ (𝐴 ≺ 1o → 𝐴 = ∅)
20		1n0 8414	. . . 4 ⊢ 1o ≠ ∅
21		1oex 8406	. . . . 5 ⊢ 1o ∈ V
22	21	0sdom 9037	. . . 4 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
23	20, 22	mpbir 232	. . 3 ⊢ ∅ ≺ 1o
24		breq1 5076	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o))
25	23, 24	mpbiri 259	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≺ 1o)
26	19, 25	impbii 210	1 ⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ≠ wne 2934  ∅c0 4262  {csn 4556   class class class wbr 5073  –1-1→wf1 6483  1_oc1o 8389   ≈ cen 8881   ≼ cdom 8882   ≺ csdm 8883

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1o 8396  df-en 8885  df-dom 8886  df-sdom 8887

This theorem is referenced by:  modom  9152  frgpcyg  21549