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Theorem ctssexmid 7031
 Description: The decidability condition in ctssdc 7005 is needed. More specifically, ctssdc 7005 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:𝑦onto𝑥) → ∃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.
StepHypRef Expression
1 ssrab2 3186 . . 3 {𝑧 ∈ ω ∣ 𝜑} ⊆ ω
2 f1oi 5412 . . . 4 ( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–1-1-onto→{𝑧 ∈ ω ∣ 𝜑}
3 f1ofo 5381 . . . 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
54elexi 2701 . . . . . . 7 ω ∈ V
65rabex 4079 . . . . . 6 {𝑧 ∈ ω ∣ 𝜑} ∈ V
7 resiexg 4871 . . . . . 6 ({𝑧 ∈ ω ∣ 𝜑} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V)
86, 7ax-mp 5 . . . . 5 ( I ↾ {𝑧 ∈ ω ∣ 𝜑}) ∈ V
9 foeq1 5348 . . . . 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}))
108, 9spcev 2783 . . . 4 (( I ↾ {𝑧 ∈ ω ∣ 𝜑}):{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑} → ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑})
112, 3, 10mp2b 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})
1312sseq1d 3130 . . . . . 6 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑦 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝜑} ⊆ ω))
14 eqidd 2141 . . . . . . . 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})
1614, 12, 15foeq123d 5368 . . . . . . 7 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:𝑦onto𝑥𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
1716exbidv 1798 . . . . . 6 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (∃𝑓 𝑓:𝑦onto𝑥 ↔ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}))
1813, 17anbi12d 465 . . . . 5 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦onto𝑥) ↔ ({𝑧 ∈ ω ∣ 𝜑} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑})))
19 djueq1 6932 . . . . . . 7 (𝑥 = {𝑧 ∈ ω ∣ 𝜑} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
20 foeq3 5350 . . . . . . 7 ((𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
2115, 19, 203syl 17 . . . . . 6 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
2221exbidv 1798 . . . . 5 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)))
2318, 22imbi12d 233 . . . 4 ((𝑥 = {𝑧 ∈ ω ∣ 𝜑} ∧ 𝑦 = {𝑧 ∈ ω ∣ 𝜑}) → (((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦onto𝑥) → ∃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:𝑦onto𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o))
256, 6, 23, 24vtocl2 2744 . . 3 (({𝑧 ∈ ω ∣ 𝜑} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝜑}–onto→{𝑧 ∈ ω ∣ 𝜑}) → ∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o))
261, 11, 25mp2an 423 . 2 𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o)
274a1i 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))
2927, 28fodjuomni 7028 . . 3 (𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
3029exlimiv 1578 . 2 (∃𝑓 𝑓:ω–onto→({𝑧 ∈ ω ∣ 𝜑} ⊔ 1o) → (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅))
31 biidd 171 . . . . . 6 (𝑧 = 𝑤 → (𝜑𝜑))
3231elrab 2843 . . . . 5 (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ↔ (𝑤 ∈ ω ∧ 𝜑))
3332simprbi 273 . . . 4 (𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
3433exlimiv 1578 . . 3 (∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} → 𝜑)
35 rabeq0 3396 . . . 4 ({𝑧 ∈ ω ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ ω ¬ 𝜑)
36 peano1 4515 . . . . 5 ∅ ∈ ω
37 elex2 2705 . . . . 5 (∅ ∈ ω → ∃𝑢 𝑢 ∈ ω)
38 r19.3rmv 3457 . . . . 5 (∃𝑢 𝑢 ∈ ω → (¬ 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑))
3936, 37, 38mp2b 8 . . . 4 𝜑 ↔ ∀𝑧 ∈ ω ¬ 𝜑)
4035, 39sylbb2 137 . . 3 ({𝑧 ∈ ω ∣ 𝜑} = ∅ → ¬ 𝜑)
4134, 40orim12i 749 . 2 ((∃𝑤 𝑤 ∈ {𝑧 ∈ ω ∣ 𝜑} ∨ {𝑧 ∈ ω ∣ 𝜑} = ∅) → (𝜑 ∨ ¬ 𝜑))
4226, 30, 41mp2b 8 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  {crab 2421  Vcvv 2689   ⊆ wss 3075  ∅c0 3367   I cid 4217  ωcom 4511   ↾ cres 4548  –onto→wfo 5128  –1-1-onto→wf1o 5129  1oc1o 6313   ⊔ cdju 6929  Omnicomni 7011 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-1o 6320  df-2o 6321  df-map 6551  df-dju 6930  df-inl 6939  df-inr 6940  df-omni 7013 This theorem is referenced by: (None)
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