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Theorem ordtri2or2exmidlem 4370
Description: A set which is 2o if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
Assertion
Ref Expression
ordtri2or2exmidlem {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Distinct variable group:   𝜑,𝑥

Proof of Theorem ordtri2or2exmidlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 497 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3306 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
3 eleq2 2158 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 638 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 272 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 588 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7 eleq2 2158 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
87biimpac 293 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
9 velsn 3483 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
108, 9sylib 121 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
11 orc 671 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12 vex 2636 . . . . . . . . . . 11 𝑦 ∈ V
1312elpr 3487 . . . . . . . . . 10 (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
1411, 13sylibr 133 . . . . . . . . 9 (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
1510, 14syl 14 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
1615adantlr 462 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17 biidd 171 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑𝜑))
1817elrab 2785 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
1918simprbi 270 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
2019ad2antlr 474 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21 biidd 171 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜑))
2221elrab 2785 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
2316, 20, 22sylanbrc 409 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24 elrabi 2782 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25 vex 2636 . . . . . . . . 9 𝑧 ∈ V
2625elpr 3487 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
2724, 26sylib 121 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2827adantl 272 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
296, 23, 28mpjaodan 750 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3029gen2 1391 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31 dftr2 3960 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
3230, 31mpbir 145 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33 ssrab2 3121 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34 2ordpr 4368 . . 3 Ord {∅, {∅}}
35 trssord 4231 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3632, 33, 34, 35mp3an 1280 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37 pp0ex 4045 . . . 4 {∅, {∅}} ∈ V
3837rabex 4004 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
3938elon 4225 . 2 ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
4036, 39mpbir 145 1 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 667  wal 1294   = wceq 1296  wcel 1445  {crab 2374  wss 3013  c0 3302  {csn 3466  {cpr 3467  Tr wtr 3958  Ord word 4213  Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222
This theorem is referenced by:  ordtri2or2exmid  4415
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