Theorem sbthom 16566

Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 16564 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 4276	. . . . . . . . . . 11 ⊢ {∅} ∈ V
2	1	ssex 4224	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ V)
3	2	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4		omex 4689	. . . . . . . . 9 ⊢ ω ∈ V
5		djuex 7233	. . . . . . . . 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 6947	. . . . . . . . . . . 12 ⊢ ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10		domrefg 6935	. . . . . . . . . . . . . 14 ⊢ (ω ∈ V → ω ≼ ω)
11	4, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ω ≼ ω
12		djudom 7283	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
13	11, 12	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14		df1o2 6591	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
15		djueq1 7230	. . . . . . . . . . . . 13 ⊢ (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) = ({∅} ⊔ ω)
17	13, 16	breqtrrdi 4128	. . . . . . . . . . 11 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18		1onn 6683	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
19		endjusym 7286	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
20	4, 18, 19	mp2an 426	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ (1o ⊔ ω)
21		omp1eom 7285	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ ω
22	20, 21	entr3i 6957	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) ≈ ω
23		domentr 6960	. . . . . . . . . . . 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 7425	. . . . . . . . . 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 4089	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31		breq2 4090	. . . . . . . . . . 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 4089	. . . . . . . . . 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 2891	. . . . . . . 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 6952	. . . . . 6 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38		bren 6912	. . . . . 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 534	. . . . . 6 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → ω ∈ Omni)
41		simplr 528	. . . . . 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 16565	. . . . 5 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
44	39, 43	exlimddv 1945	. . . 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 1920	. 2 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47		exmid01 4286	. 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 713  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800   ⊆ wss 3198  ∅c0 3492  {csn 3667   class class class wbr 4086  EXMIDwem 4282  ωcom 4686  –1-1-onto→wf1o 5323  1_oc1o 6570   ≈ cen 6902   ≼ cdom 6903   ⊔ cdju 7227  Omnicomni 7324

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-exmid 4283  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-er 6697  df-map 6814  df-en 6905  df-dom 6906  df-dju 7228  df-inl 7237  df-inr 7238  df-case 7274  df-omni 7325

This theorem is referenced by: (None)