Theorem onintexmid 4315

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)

Assertion

Ref	Expression
onintexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3543	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2		prmg 3511	. . . . . . 7 ⊢ (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
3	2	adantr 270	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4		zfpair2 3965	. . . . . . 7 ⊢ {𝑢, 𝑣} ∈ V
5		sseq1 3020	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6		eleq2 2142	. . . . . . . . . 10 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑢, 𝑣}))
7	6	exbidv 1746	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥 ∈ 𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
8	5, 7	anbi12d 456	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9		inteq 3639	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → ∩ 𝑦 = ∩ {𝑢, 𝑣})
10		id 19	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
11	9, 10	eleq12d 2149	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
12	8, 11	imbi12d 232	. . . . . . 7 ⊢ (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13		onintexmid.onint	. . . . . . 7 ⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)
14	4, 12, 13	vtocl 2653	. . . . . 6 ⊢ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
15	1, 3, 14	syl2anc 403	. . . . 5 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16		elpri 3421	. . . . 5 ⊢ (∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣} → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
17	15, 16	syl 14	. . . 4 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
18		incom 3158	. . . . . . 7 ⊢ (𝑣 ∩ 𝑢) = (𝑢 ∩ 𝑣)
19	18	eqeq1i 2088	. . . . . 6 ⊢ ((𝑣 ∩ 𝑢) = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
20		dfss1 3170	. . . . . 6 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑣 ∩ 𝑢) = 𝑢)
21		vex 2604	. . . . . . . 8 ⊢ 𝑢 ∈ V
22		vex 2604	. . . . . . . 8 ⊢ 𝑣 ∈ V
23	21, 22	intpr 3668	. . . . . . 7 ⊢ ∩ {𝑢, 𝑣} = (𝑢 ∩ 𝑣)
24	23	eqeq1i 2088	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
25	19, 20, 24	3bitr4ri 211	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ 𝑢 ⊆ 𝑣)
26	23	eqeq1i 2088	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ (𝑢 ∩ 𝑣) = 𝑣)
27		dfss1 3170	. . . . . 6 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑣)
28	26, 27	bitr4i 185	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ 𝑣 ⊆ 𝑢)
29	25, 28	orbi12i 713	. . . 4 ⊢ ((∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣) ↔ (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
30	17, 29	sylib 120	. . 3 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
31	30	rgen2a 2417	. 2 ⊢ ∀𝑢 ∈ On ∀𝑣 ∈ On (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
32	31	ordtri2or2exmid 4314	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ∩ cin 2972   ⊆ wss 2973  {cpr 3399  ∩ cint 3636  Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by: (None)