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

Theorem sbthom 13415
 Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 13413 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 4121 . . . . . . . . . . 11 {∅} ∈ V
21ssex 4074 . . . . . . . . . 10 (𝑦 ⊆ {∅} → 𝑦 ∈ V)
32adantl 275 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4 omex 4516 . . . . . . . . 9 ω ∈ V
5 djuex 6938 . . . . . . . . 9 ((𝑦 ∈ V ∧ ω ∈ V) → (𝑦 ⊔ ω) ∈ V)
63, 4, 5sylancl 410 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ∈ V)
7 simpll 519 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω))
8 ssdomg 6681 . . . . . . . . . . . 12 ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
91, 8ax-mp 5 . . . . . . . . . . 11 (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10 domrefg 6670 . . . . . . . . . . . . . 14 (ω ∈ V → ω ≼ ω)
114, 10ax-mp 5 . . . . . . . . . . . . 13 ω ≼ ω
12 djudom 6988 . . . . . . . . . . . . 13 ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
1311, 12mpan2 422 . . . . . . . . . . . 12 (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14 df1o2 6335 . . . . . . . . . . . . 13 1o = {∅}
15 djueq1 6935 . . . . . . . . . . . . 13 (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
1614, 15ax-mp 5 . . . . . . . . . . . 12 (1o ⊔ ω) = ({∅} ⊔ ω)
1713, 16breqtrrdi 3979 . . . . . . . . . . 11 (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18 1onn 6425 . . . . . . . . . . . . . 14 1o ∈ ω
19 endjusym 6991 . . . . . . . . . . . . . 14 ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
204, 18, 19mp2an 423 . . . . . . . . . . . . 13 (ω ⊔ 1o) ≈ (1o ⊔ ω)
21 omp1eom 6990 . . . . . . . . . . . . 13 (ω ⊔ 1o) ≈ ω
2220, 21entr3i 6691 . . . . . . . . . . . 12 (1o ⊔ ω) ≈ ω
23 domentr 6694 . . . . . . . . . . . 12 (((𝑦 ⊔ ω) ≼ (1o ⊔ ω) ∧ (1o ⊔ ω) ≈ ω) → (𝑦 ⊔ ω) ≼ ω)
2422, 23mpan2 422 . . . . . . . . . . 11 ((𝑦 ⊔ ω) ≼ (1o ⊔ ω) → (𝑦 ⊔ ω) ≼ ω)
259, 17, 243syl 17 . . . . . . . . . 10 (𝑦 ⊆ {∅} → (𝑦 ⊔ ω) ≼ ω)
2625adantl 275 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ≼ ω)
27 djudomr 7096 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ω ∈ V) → ω ≼ (𝑦 ⊔ ω))
283, 4, 27sylancl 410 . . . . . . . . 9 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≼ (𝑦 ⊔ ω))
2926, 28jca 304 . . . . . . . 8 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)))
30 breq1 3941 . . . . . . . . . . 11 (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31 breq2 3942 . . . . . . . . . . 11 (𝑥 = (𝑦 ⊔ ω) → (ω ≼ 𝑥 ↔ ω ≼ (𝑦 ⊔ ω)))
3230, 31anbi12d 465 . . . . . . . . . 10 (𝑥 = (𝑦 ⊔ ω) → ((𝑥 ≼ ω ∧ ω ≼ 𝑥) ↔ ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω))))
33 breq1 3941 . . . . . . . . . 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 2777 . . . . . . . 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 6686 . . . . . 6 (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38 bren 6650 . . . . . 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 524 . . . . . 6 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → ω ∈ Omni)
41 simplr 520 . . . . . 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 13414 . . . . 5 ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
4439, 43exlimddv 1871 . . . 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 1847 . 2 ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47 exmid01 4130 . 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 698  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2690   ⊆ wss 3077  ∅c0 3369  {csn 3533   class class class wbr 3938  EXMIDwem 4127  ωcom 4513  –1-1-onto→wf1o 5131  1oc1o 6315   ≈ cen 6641   ≼ cdom 6642   ⊔ cdju 6932  Omnicomni 7014 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-exmid 4128  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-1o 6322  df-2o 6323  df-er 6438  df-map 6553  df-en 6644  df-dom 6645  df-dju 6933  df-inl 6942  df-inr 6943  df-case 6979  df-omni 7016 This theorem is referenced by: (None)
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