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Theorem sdom1 9278
Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5365, ax-un 7755. (Revised by BTernaryTau, 12-Dec-2024.)
Assertion
Ref Expression
sdom1 (𝐴 ≺ 1o𝐴 = ∅)

Proof of Theorem sdom1
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df1o2 8513 . . . . . . 7 1o = {∅}
21breq2i 5151 . . . . . 6 (𝐴 ≼ 1o𝐴 ≼ {∅})
3 brdomi 8999 . . . . . . 7 (𝐴 ≼ {∅} → ∃𝑓 𝑓:𝐴1-1→{∅})
4 f1cdmsn 7302 . . . . . . . . . 10 ((𝑓:𝐴1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
5 vex 3484 . . . . . . . . . . . . 13 𝑥 ∈ V
65ensn1 9061 . . . . . . . . . . . 12 {𝑥} ≈ 1o
7 breq1 5146 . . . . . . . . . . . 12 (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
86, 7mpbiri 258 . . . . . . . . . . 11 (𝐴 = {𝑥} → 𝐴 ≈ 1o)
98exlimiv 1930 . . . . . . . . . 10 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
104, 9syl 17 . . . . . . . . 9 ((𝑓:𝐴1-1→{∅} ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 1o)
1110expcom 413 . . . . . . . 8 (𝐴 ≠ ∅ → (𝑓:𝐴1-1→{∅} → 𝐴 ≈ 1o))
1211exlimdv 1933 . . . . . . 7 (𝐴 ≠ ∅ → (∃𝑓 𝑓:𝐴1-1→{∅} → 𝐴 ≈ 1o))
133, 12syl5 34 . . . . . 6 (𝐴 ≠ ∅ → (𝐴 ≼ {∅} → 𝐴 ≈ 1o))
142, 13biimtrid 242 . . . . 5 (𝐴 ≠ ∅ → (𝐴 ≼ 1o𝐴 ≈ 1o))
15 iman 401 . . . . 5 ((𝐴 ≼ 1o𝐴 ≈ 1o) ↔ ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
1614, 15sylib 218 . . . 4 (𝐴 ≠ ∅ → ¬ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
17 brsdom 9015 . . . 4 (𝐴 ≺ 1o ↔ (𝐴 ≼ 1o ∧ ¬ 𝐴 ≈ 1o))
1816, 17sylnibr 329 . . 3 (𝐴 ≠ ∅ → ¬ 𝐴 ≺ 1o)
1918necon4ai 2972 . 2 (𝐴 ≺ 1o𝐴 = ∅)
20 1n0 8526 . . . 4 1o ≠ ∅
21 1oex 8516 . . . . 5 1o ∈ V
22210sdom 9147 . . . 4 (∅ ≺ 1o ↔ 1o ≠ ∅)
2320, 22mpbir 231 . . 3 ∅ ≺ 1o
24 breq1 5146 . . 3 (𝐴 = ∅ → (𝐴 ≺ 1o ↔ ∅ ≺ 1o))
2523, 24mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≺ 1o)
2619, 25impbii 209 1 (𝐴 ≺ 1o𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wne 2940  c0 4333  {csn 4626   class class class wbr 5143  1-1wf1 6558  1oc1o 8499  cen 8982  cdom 8983  csdm 8984
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by:  modom  9280  frgpcyg  21592
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