Theorem sbthom 16737

Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 16735 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.

Step	Hyp	Ref	Expression
1		p0ex 4284	. . . . . . . . . . 11 ⊢ {∅} ∈ V
2	1	ssex 4231	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ V)
3	2	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4		omex 4697	. . . . . . . . 9 ⊢ ω ∈ V
5		djuex 7285	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ ω ∈ V) → (𝑦 ⊔ ω) ∈ V)
6	3, 4, 5	sylancl 413	. . . . . . . 8 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ∈ V)
7		simpll 527	. . . . . . . 8 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω))
8		ssdomg 6995	. . . . . . . . . . . 12 ⊢ ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10		domrefg 6983	. . . . . . . . . . . . . 14 ⊢ (ω ∈ V → ω ≼ ω)
11	4, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ω ≼ ω
12		djudom 7335	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
13	11, 12	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14		df1o2 6639	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
15		djueq1 7282	. . . . . . . . . . . . 13 ⊢ (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) = ({∅} ⊔ ω)
17	13, 16	breqtrrdi 4135	. . . . . . . . . . 11 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18		1onn 6731	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
19		endjusym 7338	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
20	4, 18, 19	mp2an 426	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ (1o ⊔ ω)
21		omp1eom 7337	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ ω
22	20, 21	entr3i 7005	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) ≈ ω
23		domentr 7008	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊔ ω) ≼ (1o ⊔ ω) ∧ (1o ⊔ ω) ≈ ω) → (𝑦 ⊔ ω) ≼ ω)
24	22, 23	mpan2 425	. . . . . . . . . . 11 ⊢ ((𝑦 ⊔ ω) ≼ (1o ⊔ ω) → (𝑦 ⊔ ω) ≼ ω)
25	9, 17, 24	3syl 17	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → (𝑦 ⊔ ω) ≼ ω)
26	25	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ≼ ω)
27		djudomr 7478	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ ω ∈ V) → ω ≼ (𝑦 ⊔ ω))
28	3, 4, 27	sylancl 413	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≼ (𝑦 ⊔ ω))
29	26, 28	jca 306	. . . . . . . 8 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)))
30		breq1 4096	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31		breq2 4097	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (ω ≼ 𝑥 ↔ ω ≼ (𝑦 ⊔ ω)))
32	30, 31	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ⊔ ω) → ((𝑥 ≼ ω ∧ ω ≼ 𝑥) ↔ ((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω))))
33		breq1 4096	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≈ ω ↔ (𝑦 ⊔ ω) ≈ ω))
34	32, 33	imbi12d 234	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ⊔ ω) → (((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ↔ (((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)) → (𝑦 ⊔ ω) ≈ ω)))
35	34	spcgv 2894	. . . . . . . 8 ⊢ ((𝑦 ⊔ ω) ∈ V → (∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) → (((𝑦 ⊔ ω) ≼ ω ∧ ω ≼ (𝑦 ⊔ ω)) → (𝑦 ⊔ ω) ≈ ω)))
36	6, 7, 29, 35	syl3c 63	. . . . . . 7 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 ⊔ ω) ≈ ω)
37	36	ensymd 7000	. . . . . 6 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38		bren 6960	. . . . . 6 ⊢ (ω ≈ (𝑦 ⊔ ω) ↔ ∃𝑓 𝑓:ω–1-1-onto→(𝑦 ⊔ ω))
39	37, 38	sylib 122	. . . . 5 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ∃𝑓 𝑓:ω–1-1-onto→(𝑦 ⊔ ω))
40		simpllr 536	. . . . . 6 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → ω ∈ Omni)
41		simplr 529	. . . . . 6 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → 𝑦 ⊆ {∅})
42		simpr 110	. . . . . 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))
43	40, 41, 42	sbthomlem 16736	. . . . 5 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
44	39, 43	exlimddv 1947	. . . 4 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
45	44	ex 115	. . 3 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → (𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
46	45	alrimiv 1922	. 2 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47		exmid01 4294	. 2 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
48	46, 47	sylibr 134	1 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803   ⊆ wss 3201  ∅c0 3496  {csn 3673   class class class wbr 4093  EXMIDwem 4290  ωcom 4694  –1-1-onto→wf1o 5332  1_oc1o 6618   ≈ cen 6950   ≼ cdom 6951   ⊔ cdju 7279  Omnicomni 7376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-exmid 4291  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-dju 7280  df-inl 7289  df-inr 7290  df-case 7326  df-omni 7377

This theorem is referenced by: (None)