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Theorem 1sdom 8107
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 7973.) (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 4617 . 2 (𝑎 = 𝐴 → (1𝑜𝑎 ↔ 1𝑜𝐴))
2 rexeq 3128 . . 3 (𝑎 = 𝐴 → (∃𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑥 = 𝑦))
32rexeqbi1dv 3136 . 2 (𝑎 = 𝐴 → (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
4 1onn 7664 . . . 4 1𝑜 ∈ ω
5 sucdom 8101 . . . 4 (1𝑜 ∈ ω → (1𝑜𝑎 ↔ suc 1𝑜𝑎))
64, 5ax-mp 5 . . 3 (1𝑜𝑎 ↔ suc 1𝑜𝑎)
7 df-2o 7506 . . . 4 2𝑜 = suc 1𝑜
87breq1i 4620 . . 3 (2𝑜𝑎 ↔ suc 1𝑜𝑎)
9 2dom 7973 . . . 4 (2𝑜𝑎 → ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
10 df2o3 7518 . . . . 5 2𝑜 = {∅, 1𝑜}
11 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
12 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
13 0ex 4750 . . . . . . . . . . . 12 ∅ ∈ V
144elexi 3199 . . . . . . . . . . . 12 1𝑜 ∈ V
1511, 12, 13, 14funpr 5902 . . . . . . . . . . 11 (𝑥𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
16 df-ne 2791 . . . . . . . . . . 11 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
17 1n0 7520 . . . . . . . . . . . . . . 15 1𝑜 ≠ ∅
1817necomi 2844 . . . . . . . . . . . . . 14 ∅ ≠ 1𝑜
1913, 14, 11, 12fpr 6375 . . . . . . . . . . . . . 14 (∅ ≠ 1𝑜 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦})
2018, 19ax-mp 5 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦}
21 df-f1 5852 . . . . . . . . . . . . 13 ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦} ∧ Fun {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}))
2220, 21mpbiran 952 . . . . . . . . . . . 12 ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ Fun {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
2313, 11cnvsn 5577 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
2414, 12cnvsn 5577 . . . . . . . . . . . . . . 15 {⟨1𝑜, 𝑦⟩} = {⟨𝑦, 1𝑜⟩}
2523, 24uneq12i 3743 . . . . . . . . . . . . . 14 ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
26 df-pr 4151 . . . . . . . . . . . . . . . 16 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
2726cnveqi 5257 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
28 cnvun 5497 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
2927, 28eqtri 2643 . . . . . . . . . . . . . 14 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
30 df-pr 4151 . . . . . . . . . . . . . 14 {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
3125, 29, 303eqtr4i 2653 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩}
3231funeqi 5868 . . . . . . . . . . . 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 4321 . . . . . . . . . 10 ((𝑥𝑎𝑦𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36 f1ss 6063 . . . . . . . . . 10 (({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎)
3734, 35, 36syl2an 494 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎)
38 prex 4870 . . . . . . . . . 10 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ∈ V
39 f1eq1 6053 . . . . . . . . . 10 (𝑓 = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} → (𝑓:{∅, 1𝑜}–1-1𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1𝑎))
4038, 39spcev 3286 . . . . . . . . 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 3189 . . . . . . . . 9 𝑎 ∈ V
4342brdom 7911 . . . . . . . 8 ({∅, 1𝑜} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1𝑜}–1-1𝑎)
4441, 43sylibr 224 . . . . . . 7 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {∅, 1𝑜} ≼ 𝑎)
4544expcom 451 . . . . . 6 ((𝑥𝑎𝑦𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎))
4645rexlimivv 3029 . . . . 5 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎)
4710, 46syl5eqbr 4648 . . . 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 3253 1 (𝐴𝑉 → (1𝑜𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wex 1701  wcel 1987  wne 2790  wrex 2908  cun 3553  wss 3555  c0 3891  {csn 4148  {cpr 4150  cop 4154   class class class wbr 4613  ccnv 5073  suc csuc 5684  Fun wfun 5841  wf 5843  1-1wf1 5844  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499  cdom 7897  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  unxpdomlem3  8110  frgpnabl  18199  isnzr2  19182
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