Theorem sbthom 14315

Description: Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 14313 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 4183	. . . . . . . . . . 11 ⊢ {∅} ∈ V
2	1	ssex 4135	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ V)
3	2	adantl 277	. . . . . . . . 9 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ V)
4		omex 4586	. . . . . . . . 9 ⊢ ω ∈ V
5		djuex 7032	. . . . . . . . 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 6768	. . . . . . . . . . . 12 ⊢ ({∅} ∈ V → (𝑦 ⊆ {∅} → 𝑦 ≼ {∅}))
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → 𝑦 ≼ {∅})
10		domrefg 6757	. . . . . . . . . . . . . 14 ⊢ (ω ∈ V → ω ≼ ω)
11	4, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ω ≼ ω
12		djudom 7082	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ {∅} ∧ ω ≼ ω) → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
13	11, 12	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ ({∅} ⊔ ω))
14		df1o2 6420	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
15		djueq1 7029	. . . . . . . . . . . . 13 ⊢ (1o = {∅} → (1o ⊔ ω) = ({∅} ⊔ ω))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) = ({∅} ⊔ ω)
17	13, 16	breqtrrdi 4040	. . . . . . . . . . 11 ⊢ (𝑦 ≼ {∅} → (𝑦 ⊔ ω) ≼ (1o ⊔ ω))
18		1onn 6511	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
19		endjusym 7085	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ V ∧ 1o ∈ ω) → (ω ⊔ 1o) ≈ (1o ⊔ ω))
20	4, 18, 19	mp2an 426	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ (1o ⊔ ω)
21		omp1eom 7084	. . . . . . . . . . . . 13 ⊢ (ω ⊔ 1o) ≈ ω
22	20, 21	entr3i 6778	. . . . . . . . . . . 12 ⊢ (1o ⊔ ω) ≈ ω
23		domentr 6781	. . . . . . . . . . . 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 7209	. . . . . . . . . 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 4001	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ⊔ ω) → (𝑥 ≼ ω ↔ (𝑦 ⊔ ω) ≼ ω))
31		breq2 4002	. . . . . . . . . . 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 4001	. . . . . . . . . 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 2822	. . . . . . . 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 6773	. . . . . 6 ⊢ (((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) → ω ≈ (𝑦 ⊔ ω))
38		bren 6737	. . . . . 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 14314	. . . . 5 ⊢ ((((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) ∧ 𝑦 ⊆ {∅}) ∧ 𝑓:ω–1-1-onto→(𝑦 ⊔ ω)) → (𝑦 = ∅ ∨ 𝑦 = {∅}))
44	39, 43	exlimddv 1896	. . . 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 1872	. 2 ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → ∀𝑦(𝑦 ⊆ {∅} → (𝑦 = ∅ ∨ 𝑦 = {∅})))
47		exmid01 4193	. 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 708  ∀wal 1351   = wceq 1353  ∃wex 1490   ∈ wcel 2146  Vcvv 2735   ⊆ wss 3127  ∅c0 3420  {csn 3589   class class class wbr 3998  EXMIDwem 4189  ωcom 4583  –1-1-onto→wf1o 5207  1_oc1o 6400   ≈ cen 6728   ≼ cdom 6729   ⊔ cdju 7026  Omnicomni 7122

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-exmid 4190  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-1o 6407  df-2o 6408  df-er 6525  df-map 6640  df-en 6731  df-dom 6732  df-dju 7027  df-inl 7036  df-inr 7037  df-case 7073  df-omni 7123

This theorem is referenced by: (None)