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Theorem 2dom 6285
 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 6015 . . . 4 2𝑜 = {∅, {∅}}
21breq1i 3771 . . 3 (2𝑜𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3 brdomi 6230 . . 3 ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
42, 3sylbi 114 . 2 (2𝑜𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1𝐴)
5 f1f 5092 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴𝑓:{∅, {∅}}⟶𝐴)
6 0ex 3884 . . . . . 6 ∅ ∈ V
76prid1 3476 . . . . 5 ∅ ∈ {∅, {∅}}
8 ffvelrn 5300 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
95, 7, 8sylancl 392 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘∅) ∈ 𝐴)
10 p0ex 3939 . . . . . 6 {∅} ∈ V
1110prid2 3477 . . . . 5 {∅} ∈ {∅, {∅}}
12 ffvelrn 5300 . . . . 5 ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
135, 11, 12sylancl 392 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14 0nep0 3918 . . . . . 6 ∅ ≠ {∅}
1514neii 2208 . . . . 5 ¬ ∅ = {∅}
16 f1fveq 5411 . . . . . 6 ((𝑓:{∅, {∅}}–1-1𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
177, 11, 16mpanr12 415 . . . . 5 (𝑓:{∅, {∅}}–1-1𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
1815, 17mtbiri 600 . . . 4 (𝑓:{∅, {∅}}–1-1𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19 eqeq1 2046 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
2019notbid 592 . . . . 5 (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21 eqeq2 2049 . . . . . 6 (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
2221notbid 592 . . . . 5 (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
2320, 22rspc2ev 2664 . . . 4 (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
249, 13, 18, 23syl3anc 1135 . . 3 (𝑓:{∅, {∅}}–1-1𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
2524exlimiv 1489 . 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 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ∅c0 3224  {csn 3375  {cpr 3376   class class class wbr 3764  ⟶wf 4898  –1-1→wf1 4899  ‘cfv 4902  2𝑜c2o 5995   ≼ cdom 6220 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910  df-1o 6001  df-2o 6002  df-dom 6223 This theorem is referenced by: (None)
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