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Theorem 1domsn 6615
Description: A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
Assertion
Ref Expression
1domsn {𝐴} ≼ 1o

Proof of Theorem 1domsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lt1o 6242 . . . 4 ∅ ∈ 1o
21rgenw 2441 . . 3 𝑥 ∈ {𝐴}∅ ∈ 1o
3 elsni 3484 . . . . . . 7 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 271 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝐴)
5 elsni 3484 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
65adantl 272 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑦 = 𝐴)
74, 6eqtr4d 2130 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
87a1d 22 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (∅ = ∅ → 𝑥 = 𝑦))
98rgen2a 2440 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦)
10 eqid 2095 . . . 4 (𝑥 ∈ {𝐴} ↦ ∅) = (𝑥 ∈ {𝐴} ↦ ∅)
11 eqidd 2096 . . . 4 (𝑥 = 𝑦 → ∅ = ∅)
1210, 11f1mpt 5588 . . 3 ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o ↔ (∀𝑥 ∈ {𝐴}∅ ∈ 1o ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦)))
132, 9, 12mpbir2an 891 . 2 (𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o
14 1oex 6227 . . 3 1o ∈ V
1514f1dom 6557 . 2 ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o → {𝐴} ≼ 1o)
1613, 15ax-mp 7 1 {𝐴} ≼ 1o
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  wral 2370  c0 3302  {csn 3466   class class class wbr 3867  cmpt 3921  1-1wf1 5046  1oc1o 6212  cdom 6536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1o 6219  df-dom 6539
This theorem is referenced by: (None)
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