Theorem sbthom 16823

Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 16821 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 4303	. . . . . . . . . . 11 ⊢ {∅} ∈ V
2	1	ssex 4249	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ V)
3	2	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4		omex 4717	. . . . . . . . 9 ⊢ ω ∈ V
5		djuex 7336	. . . . . . . . 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 7020	. . . . . . . . . . . 12 ⊢ ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10		domrefg 7008	. . . . . . . . . . . . . 14 ⊢ (ω ∈ V → ω ≼ ω)
11	4, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ω ≼ ω
12		djudom 7386	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
13	11, 12	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14		df1o2 6663	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
15		djueq1 7333	. . . . . . . . . . . . 13 ⊢ (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) = ({∅} ⊔ ω)
17	13, 16	breqtrrdi 4153	. . . . . . . . . . 11 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18		1onn 6755	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
19		endjusym 7389	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
20	4, 18, 19	mp2an 426	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ (1o ⊔ ω)
21		omp1eom 7388	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ ω
22	20, 21	entr3i 7030	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) ≈ ω
23		domentr 7033	. . . . . . . . . . . 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 7529	. . . . . . . . . 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 4114	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31		breq2 4115	. . . . . . . . . . 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 4114	. . . . . . . . . 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 2906	. . . . . . . 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 7025	. . . . . 6 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38		bren 6985	. . . . . 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 16822	. . . . 5 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
44	39, 43	exlimddv 1950	. . . 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 1923	. 2 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47		exmid01 4313	. 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 2205  Vcvv 2815   ⊆ wss 3213  ∅c0 3510  {csn 3691   class class class wbr 4111  EXMIDwem 4309  ωcom 4714  –1-1-onto→wf1o 5353  1_oc1o 6642   ≈ cen 6975   ≼ cdom 6976   ⊔ cdju 7330  Omnicomni 7427

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-exmid 4310  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-dju 7331  df-inl 7340  df-inr 7341  df-case 7377  df-omni 7428

This theorem is referenced by: (None)