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Theorem 1sdom 8709
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8569.) (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
1sdom (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdom
Dummy variables 𝑓 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5037 . 2 (𝑎 = 𝐴 → (1o𝑎 ↔ 1o𝐴))
2 rexeq 3362 . . 3 (𝑎 = 𝐴 → (∃𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑥 = 𝑦))
32rexeqbi1dv 3360 . 2 (𝑎 = 𝐴 → (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
4 1onn 8252 . . . 4 1o ∈ ω
5 sucdom 8703 . . . 4 (1o ∈ ω → (1o𝑎 ↔ suc 1o𝑎))
64, 5ax-mp 5 . . 3 (1o𝑎 ↔ suc 1o𝑎)
7 df-2o 8090 . . . 4 2o = suc 1o
87breq1i 5040 . . 3 (2o𝑎 ↔ suc 1o𝑎)
9 2dom 8569 . . . 4 (2o𝑎 → ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
10 df2o3 8104 . . . . 5 2o = {∅, 1o}
11 vex 3447 . . . . . . . . . . . 12 𝑥 ∈ V
12 vex 3447 . . . . . . . . . . . 12 𝑦 ∈ V
13 0ex 5178 . . . . . . . . . . . 12 ∅ ∈ V
14 1oex 8097 . . . . . . . . . . . 12 1o ∈ V
1511, 12, 13, 14funpr 6384 . . . . . . . . . . 11 (𝑥𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16 df-ne 2991 . . . . . . . . . . 11 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
17 1n0 8106 . . . . . . . . . . . . . . 15 1o ≠ ∅
1817necomi 3044 . . . . . . . . . . . . . 14 ∅ ≠ 1o
1913, 14, 11, 12fpr 6897 . . . . . . . . . . . . . 14 (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
2018, 19ax-mp 5 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21 df-f1 6333 . . . . . . . . . . . . 13 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦} ∧ Fun {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
2220, 21mpbiran 708 . . . . . . . . . . . 12 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ Fun {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
2313, 11cnvsn 6054 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
2414, 12cnvsn 6054 . . . . . . . . . . . . . . 15 {⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
2523, 24uneq12i 4091 . . . . . . . . . . . . . 14 ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26 df-pr 4531 . . . . . . . . . . . . . . . 16 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
2726cnveqi 5713 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28 cnvun 5972 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
2927, 28eqtri 2824 . . . . . . . . . . . . . 14 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
30 df-pr 4531 . . . . . . . . . . . . . 14 {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
3125, 29, 303eqtr4i 2834 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
3231funeqi 6349 . . . . . . . . . . . 12 (Fun {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
3322, 32bitr2i 279 . . . . . . . . . . 11 (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
3415, 16, 333imtr3i 294 . . . . . . . . . 10 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
35 prssi 4717 . . . . . . . . . 10 ((𝑥𝑎𝑦𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36 f1ss 6559 . . . . . . . . . 10 (({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎)
3734, 35, 36syl2an 598 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎)
38 prex 5301 . . . . . . . . . 10 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39 f1eq1 6548 . . . . . . . . . 10 (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎))
4038, 39spcev 3558 . . . . . . . . 9 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎 → ∃𝑓 𝑓:{∅, 1o}–1-1𝑎)
4137, 40syl 17 . . . . . . . 8 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → ∃𝑓 𝑓:{∅, 1o}–1-1𝑎)
42 vex 3447 . . . . . . . . 9 𝑎 ∈ V
4342brdom 8508 . . . . . . . 8 ({∅, 1o} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1o}–1-1𝑎)
4441, 43sylibr 237 . . . . . . 7 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {∅, 1o} ≼ 𝑎)
4544expcom 417 . . . . . 6 ((𝑥𝑎𝑦𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎))
4645rexlimivv 3254 . . . . 5 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
4710, 46eqbrtrid 5068 . . . 4 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → 2o𝑎)
489, 47impbii 212 . . 3 (2o𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
496, 8, 483bitr2i 302 . 2 (1o𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
501, 3, 49vtoclbg 3520 1 (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wex 1781  wcel 2112  wne 2990  wrex 3110  cun 3882  wss 3884  c0 4246  {csn 4528  {cpr 4530  cop 4534   class class class wbr 5033  ccnv 5522  suc csuc 6165  Fun wfun 6322  wf 6324  1-1wf1 6325  ωcom 7564  1oc1o 8082  2oc2o 8083  cdom 8494  csdm 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-2o 8090  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499
This theorem is referenced by:  unxpdomlem3  8712  frgpnabl  18992  isnzr2  20033
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