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Theorem 2dom 9030
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 8475 . . . 4 2o = {∅, {∅}}
21breq1i 5156 . . 3 (2o𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3 brdomi 8954 . . 3 ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
42, 3sylbi 216 . 2 (2o𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
5 f1f 6788 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴𝑓:{∅, {∅}}⟶𝐴)
6 0ex 5308 . . . . . 6 ∅ ∈ V
76prid1 4767 . . . . 5 ∅ ∈ {∅, {∅}}
8 ffvelcdm 7084 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
95, 7, 8sylancl 587 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘∅) ∈ 𝐴)
10 snex 5432 . . . . . 6 {∅} ∈ V
1110prid2 4768 . . . . 5 {∅} ∈ {∅, {∅}}
12 ffvelcdm 7084 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
135, 11, 12sylancl 587 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14 0nep0 5357 . . . . . 6 ∅ ≠ {∅}
1514neii 2943 . . . . 5 ¬ ∅ = {∅}
16 f1fveq 7261 . . . . . 6 ((𝑓:{∅, {∅}}–1-1𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
177, 11, 16mpanr12 704 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
1815, 17mtbiri 327 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19 eqeq1 2737 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
2019notbid 318 . . . . 5 (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21 eqeq2 2745 . . . . . 6 (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
2221notbid 318 . . . . 5 (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
2320, 22rspc2ev 3625 . . . 4 (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
249, 13, 18, 23syl3anc 1372 . . 3 (𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
2524exlimiv 1934 . 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 205   = wceq 1542  wex 1782  wcel 2107  wrex 3071  c0 4323  {csn 4629  {cpr 4631   class class class wbr 5149  wf 6540  1-1wf1 6541  cfv 6544  2oc2o 8460  cdom 8937
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-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552  df-1o 8466  df-2o 8467  df-dom 8941
This theorem is referenced by:  1sdom  9248  1sdomOLD  9249
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