Users' Mathboxes Mathbox for Jim Kingdon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  sbthom GIF version

Theorem sbthom 12913
Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 12911 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ω? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
Assertion
Ref Expression
sbthom ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID)

Proof of Theorem sbthom
Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 p0ex 4072 . . . . . . . . . . 11 {∅} ∈ V
21ssex 4025 . . . . . . . . . 10 (𝑦 ⊆ {∅} → 𝑦 ∈ V)
32adantl 273 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4 omex 4467 . . . . . . . . 9 ω ∈ V
5 djuex 6880 . . . . . . . . 9 ((𝑦 ∈ V ∧ ω ∈ V) → (𝑦 ⊔ ω) ∈ V)
63, 4, 5sylancl 407 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ∈ V)
7 simpll 501 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω))
8 ssdomg 6626 . . . . . . . . . . . 12 ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
91, 8ax-mp 7 . . . . . . . . . . 11 (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10 domrefg 6615 . . . . . . . . . . . . . 14 (ω ∈ V → ω ≼ ω)
114, 10ax-mp 7 . . . . . . . . . . . . 13 ω ≼ ω
12 djudom 6930 . . . . . . . . . . . . 13 ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
1311, 12mpan2 419 . . . . . . . . . . . 12 (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14 df1o2 6280 . . . . . . . . . . . . 13 1o = {∅}
15 djueq1 6877 . . . . . . . . . . . . 13 (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
1614, 15ax-mp 7 . . . . . . . . . . . 12 (1o ⊔ ω) = ({∅} ⊔ ω)
1713, 16syl6breqr 3935 . . . . . . . . . . 11 (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18 1onn 6370 . . . . . . . . . . . . . 14 1o ∈ ω
19 endjusym 6933 . . . . . . . . . . . . . 14 ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
204, 18, 19mp2an 420 . . . . . . . . . . . . 13 (ω ⊔ 1o) ≈ (1o ⊔ ω)
21 omp1eom 6932 . . . . . . . . . . . . 13 (ω ⊔ 1o) ≈ ω
2220, 21entr3i 6636 . . . . . . . . . . . 12 (1o ⊔ ω) ≈ ω
23 domentr 6639 . . . . . . . . . . . 12 (((𝑦 ⊔ ω) ≼ (1o ⊔ ω) ∧ (1o ⊔ ω) ≈ ω) → (𝑦 ⊔ ω) ≼ ω)
2422, 23mpan2 419 . . . . . . . . . . 11 ((𝑦 ⊔ ω) ≼ (1o ⊔ ω) → (𝑦 ⊔ ω) ≼ ω)
259, 17, 243syl 17 . . . . . . . . . 10 (𝑦 ⊆ {∅} → (𝑦 ⊔ ω) ≼ ω)
2625adantl 273 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ≼ ω)
27 djudomr 7024 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ω ∈ V) → ω ≼ (𝑦 ⊔ ω))
283, 4, 27sylancl 407 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≼ (𝑦 ⊔ ω))
2926, 28jca 302 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)))
30 breq1 3898 . . . . . . . . . . 11 (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31 breq2 3899 . . . . . . . . . . 11 (𝑥 = (𝑦 ⊔ ω) → (ω ≼ 𝑥 ↔ ω ≼ (𝑦 ⊔ ω)))
3230, 31anbi12d 462 . . . . . . . . . 10 (𝑥 = (𝑦 ⊔ ω) → ((𝑥 ≼ ω ∧ ω ≼ 𝑥) ↔ ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω))))
33 breq1 3898 . . . . . . . . . 10 (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≈ ω ↔ (𝑦 ⊔ ω) ≈ ω))
3432, 33imbi12d 233 . . . . . . . . 9 (𝑥 = (𝑦 ⊔ ω) → (((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ↔ (((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)) → (𝑦 ⊔ ω) ≈ ω)))
3534spcgv 2744 . . . . . . . 8 ((𝑦 ⊔ ω) ∈ V → (∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) → (((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)) → (𝑦 ⊔ ω) ≈ ω)))
366, 7, 29, 35syl3c 63 . . . . . . 7 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ≈ ω)
3736ensymd 6631 . . . . . 6 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38 bren 6595 . . . . . 6 (ω ≈ (𝑦 ⊔ ω) ↔ ∃𝑓 𝑓:ω–1-1-onto→(𝑦 ⊔ ω))
3937, 38sylib 121 . . . . 5 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ∃𝑓 𝑓:ω–1-1-onto→(𝑦 ⊔ ω))
40 simpllr 506 . . . . . 6 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → ω ∈ Omni)
41 simplr 502 . . . . . 6 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → 𝑦 ⊆ {∅})
42 simpr 109 . . . . . 6 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → 𝑓:ω–1-1-onto→(𝑦 ⊔ ω))
4340, 41, 42sbthomlem 12912 . . . . 5 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
4439, 43exlimddv 1852 . . . 4 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
4544ex 114 . . 3 ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → (𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
4645alrimiv 1828 . 2 ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47 exmid01 4081 . 2 (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
4846, 47sylibr 133 1 ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680  wal 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2657  wss 3037  c0 3329  {csn 3493   class class class wbr 3895  EXMIDwem 4078  ωcom 4464  1-1-ontowf1o 5080  1oc1o 6260  cen 6586  cdom 6587  cdju 6874  Omnicomni 6954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-exmid 4079  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-1o 6267  df-2o 6268  df-er 6383  df-map 6498  df-en 6589  df-dom 6590  df-dju 6875  df-inl 6884  df-inr 6885  df-case 6921  df-omni 6956
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator