Theorem ctssexmid 7348

Description: The decidability condition in ctssdc 7311 is needed. More specifically, ctssdc 7311 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 3312	. . 3 ⊢ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω
2		f1oi 5623	. . . 4 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑}
3		f1ofo 5590	. . . 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 2815	. . . . . . 7 ⊢ ω ∈ V
6	5	rabex 4234	. . . . . 6 ⊢ {𝑧 ∈ ω ∣ 𝜑} ∈ V
7		resiexg 5058	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝜑} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V
9		foeq1 5555	. . . . 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 2901	. . . 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 110	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑦 = {𝑧 ∈ ω ∣ 𝜑})
13	12	sseq1d 3256	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑦 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω))
14		eqidd 2232	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑓 = 𝑓)
15		simpl 109	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → 𝑥 = {𝑧 ∈ ω ∣ 𝜑})
16	14, 12, 15	foeq123d 5576	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:𝑦–onto→𝑥 ↔ 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
17	16	exbidv 1873	. . . . . 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 473	. . . . 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 7238	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝜑} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
20		foeq3 5557	. . . . . . 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 1873	. . . . 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 234	. . . 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 2859	. . 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 426	. 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 7347	. . 3 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
30	29	exlimiv 1646	. 2 ⊢ (∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
31		biidd 172	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜑))
32	31	elrab 2962	. . . . 5 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ↔ (𝑤 ∈ ω ∧ 𝜑))
33	32	simprbi 275	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
34	33	exlimiv 1646	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
35		rabeq0 3524	. . . 4 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ ω ¬ 𝜑)
36		peano1 4692	. . . . 5 ⊢ ∅ ∈ ω
37		elex2 2819	. . . . 5 ⊢ (∅ ∈ ω → ∃𝑢 𝑢 ∈ ω)
38		r19.3rmv 3585	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ ω → (¬ 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑))
39	36, 37, 38	mp2b 8	. . . 4 ⊢ (¬ 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑)
40	35, 39	sylbb2 138	. . 3 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ → ¬ 𝜑)
41	34, 40	orim12i 766	. 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 104   ↔ wb 105   ∨ wo 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802   ⊆ wss 3200  ∅c0 3494   I cid 4385  ωcom 4688   ↾ cres 4727  –onto→wfo 5324  –1-1-onto→wf1o 5325  1_oc1o 6574   ⊔ cdju 7235  Omnicomni 7332

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-map 6818  df-dju 7236  df-inl 7245  df-inr 7246  df-omni 7333

This theorem is referenced by: (None)