Theorem ctssexmid 7126

Description: The decidability condition in ctssdc 7090 is needed. More specifically, ctssdc 7090 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)

Hypotheses

Ref	Expression
ctssexmid.1	⊢ ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o))
ctssexmid.lpo	⊢ ω ∈ Omni

Assertion

Ref	Expression
ctssexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑓,𝑥,𝑦

Proof of Theorem ctssexmid

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3232	. . 3 ⊢ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω
2		f1oi 5480	. . . 4 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑}
3		f1ofo 5449	. . . 4 ⊢ (( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑} → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑})
4		ctssexmid.lpo	. . . . . . . 8 ⊢ ω ∈ Omni
5	4	elexi 2742	. . . . . . 7 ⊢ ω ∈ V
6	5	rabex 4133	. . . . . 6 ⊢ {𝑧 ∈ ω ∣ 𝜑} ∈ V
7		resiexg 4936	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝜑} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V
9		foeq1 5416	. . . . 5 ⊢ (𝑓 = ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑} ↔ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
10	8, 9	spcev 2825	. . . 4 ⊢ (( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑} → ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑})
11	2, 3, 10	mp2b 8	. . 3 ⊢ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}
12		simpr 109	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑦 = {𝑧 ∈ ω ∣ 𝜑})
13	12	sseq1d 3176	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑦 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω))
14		eqidd 2171	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑓 = 𝑓)
15		simpl 108	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑥 = {𝑧 ∈ ω ∣ 𝜑})
16	14, 12, 15	foeq123d 5436	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:𝑦–onto→𝑥 ↔ 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
17	16	exbidv 1818	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (∃𝑓 𝑓:𝑦–onto→𝑥 ↔ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
18	13, 17	anbi12d 470	. . . . 5 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) ↔ ({𝑧 ∈ ω ∣ 𝜑} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑})))
19		djueq1 7017	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝜑} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
20		foeq3 5418	. . . . . . 7 ⊢ ((𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
21	15, 19, 20	3syl 17	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
22	21	exbidv 1818	. . . . 5 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
23	18, 22	imbi12d 233	. . . 4 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o)) ↔ (({𝑧 ∈ ω ∣ 𝜑} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}) → ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))))
24		ctssexmid.1	. . . 4 ⊢ ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o))
25	6, 6, 23, 24	vtocl2 2785	. . 3 ⊢ (({𝑧 ∈ ω ∣ 𝜑} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}) → ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
26	1, 11, 25	mp2an 424	. 2 ⊢ ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)
27	4	a1i 9	. . . 4 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → ω ∈ Omni)
28		id 19	. . . 4 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
29	27, 28	fodjuomni 7125	. . 3 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
30	29	exlimiv 1591	. 2 ⊢ (∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
31		biidd 171	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜑))
32	31	elrab 2886	. . . . 5 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ↔ (𝑤 ∈ ω ∧ 𝜑))
33	32	simprbi 273	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
34	33	exlimiv 1591	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
35		rabeq0 3444	. . . 4 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ ω ¬ 𝜑)
36		peano1 4578	. . . . 5 ⊢ ∅ ∈ ω
37		elex2 2746	. . . . 5 ⊢ (∅ ∈ ω → ∃𝑢 𝑢 ∈ ω)
38		r19.3rmv 3505	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ ω → (¬ 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑))
39	36, 37, 38	mp2b 8	. . . 4 ⊢ (¬ 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑)
40	35, 39	sylbb2 137	. . 3 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ → ¬ 𝜑)
41	34, 40	orim12i 754	. 2 ⊢ ((∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅) → (𝜑 ∨ ¬ 𝜑))
42	26, 30, 41	mp2b 8	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {crab 2452  Vcvv 2730   ⊆ wss 3121  ∅c0 3414   I cid 4273  ωcom 4574   ↾ cres 4613  –onto→wfo 5196  –1-1-onto→wf1o 5197  1_oc1o 6388   ⊔ cdju 7014  Omnicomni 7110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-1o 6395  df-2o 6396  df-map 6628  df-dju 7015  df-inl 7024  df-inr 7025  df-omni 7111

This theorem is referenced by: (None)