Theorem ctssexmid 7148

Description: The decidability condition in ctssdc 7112 is needed. More specifically, ctssdc 7112 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 3241	. . 3 ⊢ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω
2		f1oi 5500	. . . 4 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑}
3		f1ofo 5469	. . . 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 2750	. . . . . . 7 ⊢ ω ∈ V
6	5	rabex 4148	. . . . . 6 ⊢ {𝑧 ∈ ω ∣ 𝜑} ∈ V
7		resiexg 4953	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝜑} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V
9		foeq1 5435	. . . . 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 2833	. . . 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 3185	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑦 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω))
14		eqidd 2178	. . . . . . . 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 5455	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:𝑦–onto→𝑥 ↔ 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
17	16	exbidv 1825	. . . . . 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 7039	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝜑} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
20		foeq3 5437	. . . . . . 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 1825	. . . . 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 2793	. . 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 7147	. . 3 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
30	29	exlimiv 1598	. 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 2894	. . . . 5 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ↔ (𝑤 ∈ ω ∧ 𝜑))
33	32	simprbi 275	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
34	33	exlimiv 1598	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
35		rabeq0 3453	. . . 4 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ ω ¬ 𝜑)
36		peano1 4594	. . . . 5 ⊢ ∅ ∈ ω
37		elex2 2754	. . . . 5 ⊢ (∅ ∈ ω → ∃𝑢 𝑢 ∈ ω)
38		r19.3rmv 3514	. . . . 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 759	. 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 708   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  {crab 2459  Vcvv 2738   ⊆ wss 3130  ∅c0 3423   I cid 4289  ωcom 4590   ↾ cres 4629  –onto→wfo 5215  –1-1-onto→wf1o 5216  1_oc1o 6410   ⊔ cdju 7036  Omnicomni 7132

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-1o 6417  df-2o 6418  df-map 6650  df-dju 7037  df-inl 7046  df-inr 7047  df-omni 7133

This theorem is referenced by: (None)