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Theorem 2dom 6315
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 6045 . . . 4 2𝑜 = {∅, {∅}}
21breq1i 3798 . . 3 (2𝑜𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3 brdomi 6260 . . 3 ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
42, 3sylbi 118 . 2 (2𝑜𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
5 f1f 5119 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴𝑓:{∅, {∅}}⟶𝐴)
6 0ex 3911 . . . . . 6 ∅ ∈ V
76prid1 3503 . . . . 5 ∅ ∈ {∅, {∅}}
8 ffvelrn 5327 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
95, 7, 8sylancl 398 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘∅) ∈ 𝐴)
10 p0ex 3966 . . . . . 6 {∅} ∈ V
1110prid2 3504 . . . . 5 {∅} ∈ {∅, {∅}}
12 ffvelrn 5327 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
135, 11, 12sylancl 398 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14 0nep0 3945 . . . . . 6 ∅ ≠ {∅}
1514neii 2222 . . . . 5 ¬ ∅ = {∅}
16 f1fveq 5438 . . . . . 6 ((𝑓:{∅, {∅}}–1-1𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
177, 11, 16mpanr12 423 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
1815, 17mtbiri 610 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19 eqeq1 2062 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
2019notbid 602 . . . . 5 (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21 eqeq2 2065 . . . . . 6 (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
2221notbid 602 . . . . 5 (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
2320, 22rspc2ev 2686 . . . 4 (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
249, 13, 18, 23syl3anc 1146 . . 3 (𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
2524exlimiv 1505 . 2 (∃𝑓 𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
264, 25syl 14 1 (2𝑜𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wex 1397  wcel 1409  wrex 2324  c0 3251  {csn 3402  {cpr 3403   class class class wbr 3791  wf 4925  1-1wf1 4926  cfv 4929  2𝑜c2o 6025  cdom 6250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fv 4937  df-1o 6031  df-2o 6032  df-dom 6253
This theorem is referenced by: (None)
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