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Theorem 1sdomOLD 9246
Description: Obsolete version of 1sdom 9245 as of 30-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
1sdomOLD (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdomOLD
Dummy variables 𝑓 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5152 . 2 (𝑎 = 𝐴 → (1o𝑎 ↔ 1o𝐴))
2 rexeq 3322 . . 3 (𝑎 = 𝐴 → (∃𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑥 = 𝑦))
32rexeqbi1dv 3335 . 2 (𝑎 = 𝐴 → (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
4 1onn 8636 . . . 4 1o ∈ ω
5 sucdom 9232 . . . 4 (1o ∈ ω → (1o𝑎 ↔ suc 1o𝑎))
64, 5ax-mp 5 . . 3 (1o𝑎 ↔ suc 1o𝑎)
7 df-2o 8464 . . . 4 2o = suc 1o
87breq1i 5155 . . 3 (2o𝑎 ↔ suc 1o𝑎)
9 2dom 9027 . . . 4 (2o𝑎 → ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
10 df2o3 8471 . . . . 5 2o = {∅, 1o}
11 vex 3479 . . . . . . . . . . . 12 𝑥 ∈ V
12 vex 3479 . . . . . . . . . . . 12 𝑦 ∈ V
13 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
14 1oex 8473 . . . . . . . . . . . 12 1o ∈ V
1511, 12, 13, 14funpr 6602 . . . . . . . . . . 11 (𝑥𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16 df-ne 2942 . . . . . . . . . . 11 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
17 1n0 8485 . . . . . . . . . . . . . . 15 1o ≠ ∅
1817necomi 2996 . . . . . . . . . . . . . 14 ∅ ≠ 1o
1913, 14, 11, 12fpr 7149 . . . . . . . . . . . . . 14 (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
2018, 19ax-mp 5 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21 df-f1 6546 . . . . . . . . . . . . 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 6223 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
2414, 12cnvsn 6223 . . . . . . . . . . . . . . 15 {⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
2523, 24uneq12i 4161 . . . . . . . . . . . . . 14 ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26 df-pr 4631 . . . . . . . . . . . . . . . 16 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
2726cnveqi 5873 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28 cnvun 6140 . . . . . . . . . . . . . . 15 ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
2927, 28eqtri 2761 . . . . . . . . . . . . . 14 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
30 df-pr 4631 . . . . . . . . . . . . . 14 {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
3125, 29, 303eqtr4i 2771 . . . . . . . . . . . . 13 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
3231funeqi 6567 . . . . . . . . . . . 12 (Fun {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
3322, 32bitr2i 276 . . . . . . . . . . 11 (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
3415, 16, 333imtr3i 291 . . . . . . . . . 10 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
35 prssi 4824 . . . . . . . . . 10 ((𝑥𝑎𝑦𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36 f1ss 6791 . . . . . . . . . 10 (({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎)
3734, 35, 36syl2an 597 . . . . . . . . 9 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎)
38 prex 5432 . . . . . . . . . 10 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39 f1eq1 6780 . . . . . . . . . 10 (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1𝑎))
4038, 39spcev 3597 . . . . . . . . 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 3479 . . . . . . . . 9 𝑎 ∈ V
4342brdom 8953 . . . . . . . 8 ({∅, 1o} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1o}–1-1𝑎)
4441, 43sylibr 233 . . . . . . 7 ((¬ 𝑥 = 𝑦 ∧ (𝑥𝑎𝑦𝑎)) → {∅, 1o} ≼ 𝑎)
4544expcom 415 . . . . . 6 ((𝑥𝑎𝑦𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎))
4645rexlimivv 3200 . . . . 5 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
4710, 46eqbrtrid 5183 . . . 4 (∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦 → 2o𝑎)
489, 47impbii 208 . . 3 (2o𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
496, 8, 483bitr2i 299 . 2 (1o𝑎 ↔ ∃𝑥𝑎𝑦𝑎 ¬ 𝑥 = 𝑦)
501, 3, 49vtoclbg 3560 1 (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wex 1782  wcel 2107  wne 2941  wrex 3071  cun 3946  wss 3948  c0 4322  {csn 4628  {cpr 4630  cop 4634   class class class wbr 5148  ccnv 5675  suc csuc 6364  Fun wfun 6535  wf 6537  1-1wf1 6538  ωcom 7852  1oc1o 8456  2oc2o 8457  cdom 8934  csdm 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7853  df-1o 8463  df-2o 8464  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940
This theorem is referenced by: (None)
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