Theorem ctssexmid 7313

Description: The decidability condition in ctssdc 7276 is needed. More specifically, ctssdc 7276 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 3309	. . 3 ⊢ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω
2		f1oi 5610	. . . 4 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑}
3		f1ofo 5578	. . . 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 2812	. . . . . . 7 ⊢ ω ∈ V
6	5	rabex 4227	. . . . . 6 ⊢ {𝑧 ∈ ω ∣ 𝜑} ∈ V
7		resiexg 5049	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝜑} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V
9		foeq1 5543	. . . . 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 2898	. . . 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 3253	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑦 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω))
14		eqidd 2230	. . . . . . . 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 5564	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:𝑦–onto→𝑥 ↔ 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
17	16	exbidv 1871	. . . . . 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 7203	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝜑} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
20		foeq3 5545	. . . . . . 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 1871	. . . . 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 2856	. . 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 7312	. . 3 ⊢ (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
30	29	exlimiv 1644	. 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 2959	. . . . 5 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ↔ (𝑤 ∈ ω ∧ 𝜑))
33	32	simprbi 275	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
34	33	exlimiv 1644	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
35		rabeq0 3521	. . . 4 ⊢ ({𝑧 ∈ ω ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ ω ¬ 𝜑)
36		peano1 4685	. . . . 5 ⊢ ∅ ∈ ω
37		elex2 2816	. . . . 5 ⊢ (∅ ∈ ω → ∃𝑢 𝑢 ∈ ω)
38		r19.3rmv 3582	. . . . 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 764	. 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 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2799   ⊆ wss 3197  ∅c0 3491   I cid 4378  ωcom 4681   ↾ cres 4720  –onto→wfo 5315  –1-1-onto→wf1o 5316  1_oc1o 6553   ⊔ cdju 7200  Omnicomni 7297

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-1o 6560  df-2o 6561  df-map 6795  df-dju 7201  df-inl 7210  df-inr 7211  df-omni 7298

This theorem is referenced by: (None)