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Theorem pm54.43 7183
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
Assertion
Ref Expression
pm54.43 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))

Proof of Theorem pm54.43
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 6418 . . . . . . . 8 1o ∈ On
21elexi 2749 . . . . . . 7 1o ∈ V
32ensn1 6790 . . . . . 6 {1o} ≈ 1o
43ensymi 6776 . . . . 5 1o ≈ {1o}
5 entr 6778 . . . . 5 ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
64, 5mpan2 425 . . . 4 (𝐵 ≈ 1o𝐵 ≈ {1o})
71onirri 4539 . . . . . . 7 ¬ 1o ∈ 1o
8 disjsn 3653 . . . . . . 7 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
97, 8mpbir 146 . . . . . 6 (1o ∩ {1o}) = ∅
10 unen 6810 . . . . . 6 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ ((𝐴𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴𝐵) ≈ (1o ∪ {1o}))
119, 10mpanr2 438 . . . . 5 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1o ∪ {1o}))
1211ex 115 . . . 4 ((𝐴 ≈ 1o𝐵 ≈ {1o}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
136, 12sylan2 286 . . 3 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
14 df-2o 6412 . . . . 5 2o = suc 1o
15 df-suc 4368 . . . . 5 suc 1o = (1o ∪ {1o})
1614, 15eqtri 2198 . . . 4 2o = (1o ∪ {1o})
1716breq2i 4008 . . 3 ((𝐴𝐵) ≈ 2o ↔ (𝐴𝐵) ≈ (1o ∪ {1o}))
1813, 17syl6ibr 162 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2o))
19 en1 6793 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 6793 . . 3 (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21 1nen2 6855 . . . . . . . . . . . . 13 ¬ 1o ≈ 2o
2221a1i 9 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ¬ 1o ≈ 2o)
23 sneq 3602 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2423uneq2d 3289 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
25 unidm 3278 . . . . . . . . . . . . . . . 16 ({𝑥} ∪ {𝑥}) = {𝑥}
2624, 25eqtr3di 2225 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27 vex 2740 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2827ensn1 6790 . . . . . . . . . . . . . . 15 {𝑥} ≈ 1o
2926, 28eqbrtrdi 4039 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1o)
3029ensymd 6777 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → 1o ≈ ({𝑥} ∪ {𝑦}))
31 entr 6778 . . . . . . . . . . . . 13 ((1o ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
3230, 31sylan 283 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
3322, 32mtand 665 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3433necon2ai 2401 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2o𝑥𝑦)
35 disjsn2 3654 . . . . . . . . . 10 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
3634, 35syl 14 . . . . . . . . 9 (({𝑥} ∪ {𝑦}) ≈ 2o → ({𝑥} ∩ {𝑦}) = ∅)
3736a1i 9 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (({𝑥} ∪ {𝑦}) ≈ 2o → ({𝑥} ∩ {𝑦}) = ∅))
38 uneq12 3284 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3938breq1d 4010 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
40 ineq12 3331 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4140eqeq1d 2186 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4237, 39, 413imtr4d 203 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4342ex 115 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4443exlimdv 1819 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4544exlimiv 1598 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4645imp 124 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4719, 20, 46syl2anb 291 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4818, 47impbid 129 1 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wne 2347  cun 3127  cin 3128  c0 3422  {csn 3591   class class class wbr 4000  Oncon0 4360  suc csuc 4362  1oc1o 6404  2oc2o 6405  cen 6732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735
This theorem is referenced by:  pr2nelem  7184  dju1p1e2  7190
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