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Theorem onsucelsucexmidlem 4198
Description: Lemma for onsucelsucexmid 4199. The set {x {∅, {∅}} ∣ (x = ∅ φ)} appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5427), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4192. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem {x {∅, {∅}} ∣ (x = ∅ φ)} On
Distinct variable group:   φ,x

Proof of Theorem onsucelsucexmidlem
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 469 . . . . . . . 8 (((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) z = ∅) → y z)
2 noel 3205 . . . . . . . . . 10 ¬ y
3 eleq2 2083 . . . . . . . . . 10 (z = ∅ → (y zy ∅))
42, 3mtbiri 587 . . . . . . . . 9 (z = ∅ → ¬ y z)
54adantl 262 . . . . . . . 8 (((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) z = ∅) → ¬ y z)
61, 5pm2.21dd 538 . . . . . . 7 (((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) z = ∅) → y {x {∅, {∅}} ∣ (x = ∅ φ)})
76ex 108 . . . . . 6 ((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → (z = ∅ → y {x {∅, {∅}} ∣ (x = ∅ φ)}))
8 eleq2 2083 . . . . . . . . . . 11 (z = {∅} → (y zy {∅}))
98biimpac 282 . . . . . . . . . 10 ((y z z = {∅}) → y {∅})
10 elsn 3365 . . . . . . . . . 10 (y {∅} ↔ y = ∅)
119, 10sylib 127 . . . . . . . . 9 ((y z z = {∅}) → y = ∅)
12 onsucelsucexmidlem1 4197 . . . . . . . . 9 {x {∅, {∅}} ∣ (x = ∅ φ)}
1311, 12syl6eqel 2110 . . . . . . . 8 ((y z z = {∅}) → y {x {∅, {∅}} ∣ (x = ∅ φ)})
1413ex 108 . . . . . . 7 (y z → (z = {∅} → y {x {∅, {∅}} ∣ (x = ∅ φ)}))
1514adantr 261 . . . . . 6 ((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → (z = {∅} → y {x {∅, {∅}} ∣ (x = ∅ φ)}))
16 elrabi 2672 . . . . . . . 8 (z {x {∅, {∅}} ∣ (x = ∅ φ)} → z {∅, {∅}})
17 vex 2538 . . . . . . . . 9 z V
1817elpr 3368 . . . . . . . 8 (z {∅, {∅}} ↔ (z = ∅ z = {∅}))
1916, 18sylib 127 . . . . . . 7 (z {x {∅, {∅}} ∣ (x = ∅ φ)} → (z = ∅ z = {∅}))
2019adantl 262 . . . . . 6 ((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → (z = ∅ z = {∅}))
217, 15, 20mpjaod 625 . . . . 5 ((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → y {x {∅, {∅}} ∣ (x = ∅ φ)})
2221gen2 1319 . . . 4 yz((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → y {x {∅, {∅}} ∣ (x = ∅ φ)})
23 dftr2 3830 . . . 4 (Tr {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ yz((y z z {x {∅, {∅}} ∣ (x = ∅ φ)}) → y {x {∅, {∅}} ∣ (x = ∅ φ)}))
2422, 23mpbir 134 . . 3 Tr {x {∅, {∅}} ∣ (x = ∅ φ)}
25 ssrab2 3002 . . 3 {x {∅, {∅}} ∣ (x = ∅ φ)} ⊆ {∅, {∅}}
26 ord0 4077 . . . . 5 Ord ∅
27 ordsucim 4176 . . . . 5 (Ord ∅ → Ord suc ∅)
28 ordsucim 4176 . . . . 5 (Ord suc ∅ → Ord suc suc ∅)
2926, 27, 28mp2b 8 . . . 4 Ord suc suc ∅
30 df-suc 4057 . . . . . 6 suc {∅} = ({∅} ∪ {{∅}})
31 suc0 4097 . . . . . . 7 suc ∅ = {∅}
32 suceq 4088 . . . . . . 7 (suc ∅ = {∅} → suc suc ∅ = suc {∅})
3331, 32ax-mp 7 . . . . . 6 suc suc ∅ = suc {∅}
34 df-pr 3357 . . . . . 6 {∅, {∅}} = ({∅} ∪ {{∅}})
3530, 33, 343eqtr4i 2052 . . . . 5 suc suc ∅ = {∅, {∅}}
36 ordeq 4058 . . . . 5 (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
3735, 36ax-mp 7 . . . 4 (Ord suc suc ∅ ↔ Ord {∅, {∅}})
3829, 37mpbi 133 . . 3 Ord {∅, {∅}}
39 trssord 4066 . . 3 ((Tr {x {∅, {∅}} ∣ (x = ∅ φ)} {x {∅, {∅}} ∣ (x = ∅ φ)} ⊆ {∅, {∅}} Ord {∅, {∅}}) → Ord {x {∅, {∅}} ∣ (x = ∅ φ)})
4024, 25, 38, 39mp3an 1217 . 2 Ord {x {∅, {∅}} ∣ (x = ∅ φ)}
41 pp0ex 3914 . . . 4 {∅, {∅}} V
4241rabex 3875 . . 3 {x {∅, {∅}} ∣ (x = ∅ φ)} V
4342elon 4060 . 2 ({x {∅, {∅}} ∣ (x = ∅ φ)} On ↔ Ord {x {∅, {∅}} ∣ (x = ∅ φ)})
4440, 43mpbir 134 1 {x {∅, {∅}} ∣ (x = ∅ φ)} On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616  wal 1226   = wceq 1228   wcel 1374  {crab 2288  cun 2892  wss 2894  c0 3201  {csn 3350  {cpr 3351  Tr wtr 3828  Ord word 4048  Oncon0 4049  suc csuc 4051
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by:  onsucelsucexmid  4199  acexmidlemcase  5431  acexmidlemv  5434
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