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Theorem 1domsn 6761
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 6384 . . . 4 ∅ ∈ 1o
21rgenw 2512 . . 3 𝑥 ∈ {𝐴}∅ ∈ 1o
3 elsni 3578 . . . . . . 7 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 274 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝐴)
5 elsni 3578 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
65adantl 275 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑦 = 𝐴)
74, 6eqtr4d 2193 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
87a1d 22 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (∅ = ∅ → 𝑥 = 𝑦))
98rgen2a 2511 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦)
10 eqid 2157 . . . 4 (𝑥 ∈ {𝐴} ↦ ∅) = (𝑥 ∈ {𝐴} ↦ ∅)
11 eqidd 2158 . . . 4 (𝑥 = 𝑦 → ∅ = ∅)
1210, 11f1mpt 5718 . . 3 ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o ↔ (∀𝑥 ∈ {𝐴}∅ ∈ 1o ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (∅ = ∅ → 𝑥 = 𝑦)))
132, 9, 12mpbir2an 927 . 2 (𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o
14 1oex 6368 . . 3 1o ∈ V
1514f1dom 6702 . 2 ((𝑥 ∈ {𝐴} ↦ ∅):{𝐴}–1-1→1o → {𝐴} ≼ 1o)
1613, 15ax-mp 5 1 {𝐴} ≼ 1o
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wral 2435  c0 3394  {csn 3560   class class class wbr 3965  cmpt 4025  1-1wf1 5166  1oc1o 6353  cdom 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1o 6360  df-dom 6684
This theorem is referenced by: (None)
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