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Theorem 1sdom 8204
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8070.) (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 4689 . 2 (𝑎 = 𝐴 → (1𝑜𝑎 ↔ 1𝑜𝐴))
2 rexeq 3169 . . 3 (𝑎 = 𝐴 → (∃𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑥 = 𝑦))
32rexeqbi1dv 3177 . 2 (𝑎 = 𝐴 → (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
4 1onn 7764 . . . 4 1𝑜 ∈ ω
5 sucdom 8198 . . . 4 (1𝑜 ∈ ω → (1𝑜𝑎 ↔ suc 1𝑜𝑎))
64, 5ax-mp 5 . . 3 (1𝑜𝑎 ↔ suc 1𝑜𝑎)
7 df-2o 7606 . . . 4 2𝑜 = suc 1𝑜
87breq1i 4692 . . 3 (2𝑜𝑎 ↔ suc 1𝑜𝑎)
9 2dom 8070 . . . 4 (2𝑜𝑎 → ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
10 df2o3 7618 . . . . 5 2𝑜 = {∅, 1𝑜}
11 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
12 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
13 0ex 4823 . . . . . . . . . . . 12 ∅ ∈ V
144elexi 3244 . . . . . . . . . . . 12 1𝑜 ∈ V
1511, 12, 13, 14funpr 5982 . . . . . . . . . . 11 (𝑥𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
16 df-ne 2824 . . . . . . . . . . 11 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
17 1n0 7620 . . . . . . . . . . . . . . 15 1𝑜 ≠ ∅
1817necomi 2877 . . . . . . . . . . . . . 14 ∅ ≠ 1𝑜
1913, 14, 11, 12fpr 6461 . . . . . . . . . . . . . 14 (∅ ≠ 1𝑜 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦})
2018, 19ax-mp 5 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦}
21 df-f1 5931 . . . . . . . . . . . . 13 ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦} ∧ Fun {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}))
2220, 21mpbiran 973 . . . . . . . . . . . 12 ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ Fun {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
2313, 11cnvsn 5655 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
2414, 12cnvsn 5655 . . . . . . . . . . . . . . 15 {⟨1𝑜, 𝑦⟩} = {⟨𝑦, 1𝑜⟩}
2523, 24uneq12i 3798 . . . . . . . . . . . . . 14 ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
26 df-pr 4213 . . . . . . . . . . . . . . . 16 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
2726cnveqi 5329 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
28 cnvun 5573 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
2927, 28eqtri 2673 . . . . . . . . . . . . . 14 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
30 df-pr 4213 . . . . . . . . . . . . . 14 {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
3125, 29, 303eqtr4i 2683 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩}
3231funeqi 5947 . . . . . . . . . . . 12 (Fun {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
3322, 32bitr2i 265 . . . . . . . . . . 11 (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
3415, 16, 333imtr3i 280 . . . . . . . . . 10 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
35 prssi 4385 . . . . . . . . . 10 ((𝑥𝑎𝑦𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36 f1ss 6144 . . . . . . . . . 10 (({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎)
3734, 35, 36syl2an 493 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎)
38 prex 4939 . . . . . . . . . 10 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ∈ V
39 f1eq1 6134 . . . . . . . . . 10 (𝑓 = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} → (𝑓:{∅, 1𝑜}–1-1𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎))
4038, 39spcev 3331 . . . . . . . . 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 3234 . . . . . . . . 9 𝑎 ∈ V
4342brdom 8009 . . . . . . . 8 ({∅, 1𝑜} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1𝑜}–1-1𝑎)
4441, 43sylibr 224 . . . . . . 7 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {∅, 1𝑜} ≼ 𝑎)
4544expcom 450 . . . . . 6 ((𝑥𝑎𝑦𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎))
4645rexlimivv 3065 . . . . 5 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎)
4710, 46syl5eqbr 4720 . . . 4 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → 2𝑜𝑎)
489, 47impbii 199 . . 3 (2𝑜𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
496, 8, 483bitr2i 288 . 2 (1𝑜𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
501, 3, 49vtoclbg 3298 1 (𝐴𝑉 → (1𝑜𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wex 1744  wcel 2030  wne 2823  wrex 2942  cun 3605  wss 3607  c0 3948  {csn 4210  {cpr 4212  cop 4216   class class class wbr 4685  ccnv 5142  suc csuc 5763  Fun wfun 5920  wf 5922  1-1wf1 5923  ωcom 7107  1𝑜c1o 7598  2𝑜c2o 7599  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  unxpdomlem3  8207  frgpnabl  18324  isnzr2  19311
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