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Theorem 1sdom 9248
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 9030.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7725. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
1sdom (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdom
StepHypRef Expression
1 1sdom2dom 9247 . 2 (1o𝐴 ↔ 2o𝐴)
2 2dom 9030 . . 3 (2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
3 df-ne 2942 . . . . . 6 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
432rexbii 3130 . . . . 5 (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
5 rex2dom 9246 . . . . 5 ((𝐴𝑉 ∧ ∃𝑥𝐴𝑦𝐴 𝑥𝑦) → 2o𝐴)
64, 5sylan2br 596 . . . 4 ((𝐴𝑉 ∧ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦) → 2o𝐴)
76ex 414 . . 3 (𝐴𝑉 → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦 → 2o𝐴))
82, 7impbid2 225 . 2 (𝐴𝑉 → (2o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
91, 8bitrid 283 1 (𝐴𝑉 → (1o𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2107  wne 2941  wrex 3071   class class class wbr 5149  1oc1o 8459  2oc2o 8460  cdom 8937  csdm 8938
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 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  unxpdomlem3  9252  frgpnabl  19743  isnzr2  20297
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