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Theorem ordtri2or2exmidlem 4251
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 481 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3228 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
3 eleq2 2101 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 600 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 262 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 550 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7 eleq2 2101 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
87biimpac 282 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
9 velsn 3392 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
108, 9sylib 127 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
11 orc 633 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12 vex 2560 . . . . . . . . . . 11 𝑦 ∈ V
1312elpr 3396 . . . . . . . . . 10 (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
1411, 13sylibr 137 . . . . . . . . 9 (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
1510, 14syl 14 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
1615adantlr 446 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17 biidd 161 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑𝜑))
1817elrab 2698 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
1918simprbi 260 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
2019ad2antlr 458 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21 biidd 161 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜑))
2221elrab 2698 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
2316, 20, 22sylanbrc 394 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24 elrabi 2695 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25 vex 2560 . . . . . . . . 9 𝑧 ∈ V
2625elpr 3396 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
2724, 26sylib 127 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2827adantl 262 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
296, 23, 28mpjaodan 711 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3029gen2 1339 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31 dftr2 3856 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
3230, 31mpbir 134 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33 ssrab2 3025 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34 2ordpr 4249 . . 3 Ord {∅, {∅}}
35 trssord 4117 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3632, 33, 34, 35mp3an 1232 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37 pp0ex 3940 . . . 4 {∅, {∅}} ∈ V
3837rabex 3901 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
3938elon 4111 . 2 ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
4036, 39mpbir 134 1 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wo 629  wal 1241   = wceq 1243  wcel 1393  {crab 2310  wss 2917  c0 3224  {csn 3375  {cpr 3376  Tr wtr 3854  Ord word 4099  Oncon0 4100
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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927
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-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108
This theorem is referenced by:  ordtri2or2exmid  4296
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