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Theorem 1sdom 9199
Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8981.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7677. (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 9198 . 2 (1o𝐴 ↔ 2o𝐴)
2 2dom 8981 . . 3 (2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
3 df-ne 2945 . . . . . 6 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
432rexbii 3129 . . . . 5 (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
5 rex2dom 9197 . . . . 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 2944  wrex 3074   class class class wbr 5110  1oc1o 8410  2oc2o 8411  cdom 8888  csdm 8889
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1o 8417  df-2o 8418  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by:  unxpdomlem3  9203  frgpnabl  19660  isnzr2  20749
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