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Theorem onintexmid 4267
Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis is an instance of ((𝐴 ⊆ On ∧ ∃𝑥𝑥𝐴) → 𝐴𝐴) which we would be able to prove given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint (({𝑦, 𝑧} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑦, 𝑧}) → {𝑦, 𝑧} ∈ {𝑦, 𝑧})
Assertion
Ref Expression
onintexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable groups:   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem onintexmid
StepHypRef Expression
1 prssi 3519 . . . . . 6 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → {𝑦, 𝑧} ⊆ On)
2 prmg 3486 . . . . . . 7 (𝑦 ∈ On → ∃𝑥 𝑥 ∈ {𝑦, 𝑧})
32adantr 261 . . . . . 6 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ∃𝑥 𝑥 ∈ {𝑦, 𝑧})
4 onintexmid.onint . . . . . 6 (({𝑦, 𝑧} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑦, 𝑧}) → {𝑦, 𝑧} ∈ {𝑦, 𝑧})
51, 3, 4syl2anc 391 . . . . 5 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → {𝑦, 𝑧} ∈ {𝑦, 𝑧})
6 elpri 3395 . . . . 5 ( {𝑦, 𝑧} ∈ {𝑦, 𝑧} → ( {𝑦, 𝑧} = 𝑦 {𝑦, 𝑧} = 𝑧))
75, 6syl 14 . . . 4 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ( {𝑦, 𝑧} = 𝑦 {𝑦, 𝑧} = 𝑧))
8 incom 3126 . . . . . . 7 (𝑧𝑦) = (𝑦𝑧)
98eqeq1i 2047 . . . . . 6 ((𝑧𝑦) = 𝑦 ↔ (𝑦𝑧) = 𝑦)
10 dfss1 3138 . . . . . 6 (𝑦𝑧 ↔ (𝑧𝑦) = 𝑦)
11 vex 2557 . . . . . . . 8 𝑦 ∈ V
12 vex 2557 . . . . . . . 8 𝑧 ∈ V
1311, 12intpr 3644 . . . . . . 7 {𝑦, 𝑧} = (𝑦𝑧)
1413eqeq1i 2047 . . . . . 6 ( {𝑦, 𝑧} = 𝑦 ↔ (𝑦𝑧) = 𝑦)
159, 10, 143bitr4ri 202 . . . . 5 ( {𝑦, 𝑧} = 𝑦𝑦𝑧)
1613eqeq1i 2047 . . . . . 6 ( {𝑦, 𝑧} = 𝑧 ↔ (𝑦𝑧) = 𝑧)
17 dfss1 3138 . . . . . 6 (𝑧𝑦 ↔ (𝑦𝑧) = 𝑧)
1816, 17bitr4i 176 . . . . 5 ( {𝑦, 𝑧} = 𝑧𝑧𝑦)
1915, 18orbi12i 681 . . . 4 (( {𝑦, 𝑧} = 𝑦 {𝑦, 𝑧} = 𝑧) ↔ (𝑦𝑧𝑧𝑦))
207, 19sylib 127 . . 3 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦𝑧𝑧𝑦))
2120rgen2a 2372 . 2 𝑦 ∈ On ∀𝑧 ∈ On (𝑦𝑧𝑧𝑦)
2221ordtri2or2exmid 4266 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wo 629   = wceq 1243  wex 1381  wcel 1393  cin 2913  wss 2914  {cpr 3373   cint 3612  Oncon0 4072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3578  df-int 3613  df-tr 3852  df-iord 4075  df-on 4077  df-suc 4080
This theorem is referenced by: (None)
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