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Theorem 2dom 6221
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
Assertion
Ref Expression
2dom (2𝑜Ax A y A ¬ 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 5954 . . . 4 2𝑜 = {∅, {∅}}
21breq1i 3762 . . 3 (2𝑜A ↔ {∅, {∅}} ≼ A)
3 brdomi 6166 . . 3 ({∅, {∅}} ≼ Af f:{∅, {∅}}–1-1A)
42, 3sylbi 114 . 2 (2𝑜Af f:{∅, {∅}}–1-1A)
5 f1f 5035 . . . . 5 (f:{∅, {∅}}–1-1Af:{∅, {∅}}⟶A)
6 0ex 3875 . . . . . 6 V
76prid1 3467 . . . . 5 {∅, {∅}}
8 ffvelrn 5243 . . . . 5 ((f:{∅, {∅}}⟶A {∅, {∅}}) → (f‘∅) A)
95, 7, 8sylancl 392 . . . 4 (f:{∅, {∅}}–1-1A → (f‘∅) A)
10 p0ex 3930 . . . . . 6 {∅} V
1110prid2 3468 . . . . 5 {∅} {∅, {∅}}
12 ffvelrn 5243 . . . . 5 ((f:{∅, {∅}}⟶A {∅} {∅, {∅}}) → (f‘{∅}) A)
135, 11, 12sylancl 392 . . . 4 (f:{∅, {∅}}–1-1A → (f‘{∅}) A)
14 0nep0 3909 . . . . . 6 ∅ ≠ {∅}
1514neii 2205 . . . . 5 ¬ ∅ = {∅}
16 f1fveq 5354 . . . . . 6 ((f:{∅, {∅}}–1-1A (∅ {∅, {∅}} {∅} {∅, {∅}})) → ((f‘∅) = (f‘{∅}) ↔ ∅ = {∅}))
177, 11, 16mpanr12 415 . . . . 5 (f:{∅, {∅}}–1-1A → ((f‘∅) = (f‘{∅}) ↔ ∅ = {∅}))
1815, 17mtbiri 599 . . . 4 (f:{∅, {∅}}–1-1A → ¬ (f‘∅) = (f‘{∅}))
19 eqeq1 2043 . . . . . 6 (x = (f‘∅) → (x = y ↔ (f‘∅) = y))
2019notbid 591 . . . . 5 (x = (f‘∅) → (¬ x = y ↔ ¬ (f‘∅) = y))
21 eqeq2 2046 . . . . . 6 (y = (f‘{∅}) → ((f‘∅) = y ↔ (f‘∅) = (f‘{∅})))
2221notbid 591 . . . . 5 (y = (f‘{∅}) → (¬ (f‘∅) = y ↔ ¬ (f‘∅) = (f‘{∅})))
2320, 22rspc2ev 2658 . . . 4 (((f‘∅) A (f‘{∅}) A ¬ (f‘∅) = (f‘{∅})) → x A y A ¬ x = y)
249, 13, 18, 23syl3anc 1134 . . 3 (f:{∅, {∅}}–1-1Ax A y A ¬ x = y)
2524exlimiv 1486 . 2 (f f:{∅, {∅}}–1-1Ax A y A ¬ x = y)
264, 25syl 14 1 (2𝑜Ax A y A ¬ x = y)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301  c0 3218  {csn 3367  {cpr 3368   class class class wbr 3755  wf 4841  1-1wf1 4842  cfv 4845  2𝑜c2o 5934  cdom 6156
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853  df-1o 5940  df-2o 5941  df-dom 6159
This theorem is referenced by: (None)
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