Theorem sbthom 16106

Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 16104 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 4240	. . . . . . . . . . 11 ⊢ {∅} ∈ V
2	1	ssex 4189	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ V)
3	2	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4		omex 4649	. . . . . . . . 9 ⊢ ω ∈ V
5		djuex 7160	. . . . . . . . 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 6883	. . . . . . . . . . . 12 ⊢ ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10		domrefg 6871	. . . . . . . . . . . . . 14 ⊢ (ω ∈ V → ω ≼ ω)
11	4, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ω ≼ ω
12		djudom 7210	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
13	11, 12	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14		df1o2 6528	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
15		djueq1 7157	. . . . . . . . . . . . 13 ⊢ (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) = ({∅} ⊔ ω)
17	13, 16	breqtrrdi 4093	. . . . . . . . . . 11 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18		1onn 6619	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
19		endjusym 7213	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
20	4, 18, 19	mp2an 426	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ (1o ⊔ ω)
21		omp1eom 7212	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ ω
22	20, 21	entr3i 6893	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) ≈ ω
23		domentr 6896	. . . . . . . . . . . 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 7348	. . . . . . . . . 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 4054	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31		breq2 4055	. . . . . . . . . . 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 4054	. . . . . . . . . 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 2864	. . . . . . . 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 6888	. . . . . 6 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38		bren 6848	. . . . . 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 16105	. . . . 5 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
44	39, 43	exlimddv 1923	. . . 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 1898	. 2 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47		exmid01 4250	. 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 710  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773   ⊆ wss 3170  ∅c0 3464  {csn 3638   class class class wbr 4051  EXMIDwem 4246  ωcom 4646  –1-1-onto→wf1o 5279  1_oc1o 6508   ≈ cen 6838   ≼ cdom 6839   ⊔ cdju 7154  Omnicomni 7251

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-exmid 4247  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-1o 6515  df-2o 6516  df-er 6633  df-map 6750  df-en 6841  df-dom 6842  df-dju 7155  df-inl 7164  df-inr 7165  df-case 7201  df-omni 7252

This theorem is referenced by: (None)