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Theorem 2dom 9070
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
Assertion
Ref Expression
2dom (2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem 2dom
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df2o2 8515 . . . 4 2o = {∅, {∅}}
21breq1i 5150 . . 3 (2o𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3 brdomi 8999 . . 3 ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
42, 3sylbi 217 . 2 (2o𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
5 f1f 6804 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴𝑓:{∅, {∅}}⟶𝐴)
6 0ex 5307 . . . . . 6 ∅ ∈ V
76prid1 4762 . . . . 5 ∅ ∈ {∅, {∅}}
8 ffvelcdm 7101 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
95, 7, 8sylancl 586 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘∅) ∈ 𝐴)
10 snex 5436 . . . . . 6 {∅} ∈ V
1110prid2 4763 . . . . 5 {∅} ∈ {∅, {∅}}
12 ffvelcdm 7101 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
135, 11, 12sylancl 586 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14 0nep0 5358 . . . . . 6 ∅ ≠ {∅}
1514neii 2942 . . . . 5 ¬ ∅ = {∅}
16 f1fveq 7282 . . . . . 6 ((𝑓:{∅, {∅}}–1-1𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
177, 11, 16mpanr12 705 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
1815, 17mtbiri 327 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19 eqeq1 2741 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
2019notbid 318 . . . . 5 (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21 eqeq2 2749 . . . . . 6 (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
2221notbid 318 . . . . 5 (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
2320, 22rspc2ev 3635 . . . 4 (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
249, 13, 18, 23syl3anc 1373 . . 3 (𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
2524exlimiv 1930 . 2 (∃𝑓 𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
264, 25syl 17 1 (2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wex 1779  wcel 2108  wrex 3070  c0 4333  {csn 4626  {cpr 4628   class class class wbr 5143  wf 6557  1-1wf1 6558  cfv 6561  2oc2o 8500  cdom 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-1o 8506  df-2o 8507  df-dom 8987
This theorem is referenced by:  1sdom  9284  1sdomOLD  9285
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