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Theorem pm54.43 6797
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 6170 . . . . . . . 8 1𝑜 ∈ On
21elexi 2631 . . . . . . 7 1𝑜 ∈ V
32ensn1 6493 . . . . . 6 {1𝑜} ≈ 1𝑜
43ensymi 6479 . . . . 5 1𝑜 ≈ {1𝑜}
5 entr 6481 . . . . 5 ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
64, 5mpan2 416 . . . 4 (𝐵 ≈ 1𝑜𝐵 ≈ {1𝑜})
71onirri 4349 . . . . . . 7 ¬ 1𝑜 ∈ 1𝑜
8 disjsn 3499 . . . . . . 7 ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
97, 8mpbir 144 . . . . . 6 (1𝑜 ∩ {1𝑜}) = ∅
10 unen 6513 . . . . . 6 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ ((𝐴𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
119, 10mpanr2 429 . . . . 5 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1211ex 113 . . . 4 ((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
136, 12sylan2 280 . . 3 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14 df-2o 6164 . . . . 5 2𝑜 = suc 1𝑜
15 df-suc 4189 . . . . 5 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
1614, 15eqtri 2108 . . . 4 2𝑜 = (1𝑜 ∪ {1𝑜})
1716breq2i 3845 . . 3 ((𝐴𝐵) ≈ 2𝑜 ↔ (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1813, 17syl6ibr 160 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2𝑜))
19 en1 6496 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 6496 . . 3 (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21 1nen2 6557 . . . . . . . . . . . . 13 ¬ 1𝑜 ≈ 2𝑜
2221a1i 9 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ¬ 1𝑜 ≈ 2𝑜)
23 unidm 3141 . . . . . . . . . . . . . . . 16 ({𝑥} ∪ {𝑥}) = {𝑥}
24 sneq 3452 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2524uneq2d 3152 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2623, 25syl5reqr 2135 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27 vex 2622 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2827ensn1 6493 . . . . . . . . . . . . . . 15 {𝑥} ≈ 1𝑜
2926, 28syl6eqbr 3874 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1𝑜)
3029ensymd 6480 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → 1𝑜 ≈ ({𝑥} ∪ {𝑦}))
31 entr 6481 . . . . . . . . . . . . 13 ((1𝑜 ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
3230, 31sylan 277 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
3322, 32mtand 626 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3433necon2ai 2309 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2𝑜𝑥𝑦)
35 disjsn2 3500 . . . . . . . . . 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 3147 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3938breq1d 3847 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
40 ineq12 3194 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4140eqeq1d 2096 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4237, 39, 413imtr4d 201 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4342ex 113 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4443exlimdv 1747 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4544exlimiv 1534 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4645imp 122 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4719, 20, 46syl2anb 285 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4818, 47impbid 127 1 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wne 2255  cun 2995  cin 2996  c0 3284  {csn 3441   class class class wbr 3837  Oncon0 4181  suc csuc 4183  1𝑜c1o 6156  2𝑜c2o 6157  cen 6435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1o 6163  df-2o 6164  df-er 6272  df-en 6438
This theorem is referenced by:  pr2nelem  6798  dju1p1e2  6802
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