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Theorem pm54.43 7119
 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 6367 . . . . . . . 8 1o ∈ On
21elexi 2724 . . . . . . 7 1o ∈ V
32ensn1 6738 . . . . . 6 {1o} ≈ 1o
43ensymi 6724 . . . . 5 1o ≈ {1o}
5 entr 6726 . . . . 5 ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
64, 5mpan2 422 . . . 4 (𝐵 ≈ 1o𝐵 ≈ {1o})
71onirri 4501 . . . . . . 7 ¬ 1o ∈ 1o
8 disjsn 3621 . . . . . . 7 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
97, 8mpbir 145 . . . . . 6 (1o ∩ {1o}) = ∅
10 unen 6758 . . . . . 6 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ ((𝐴𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴𝐵) ≈ (1o ∪ {1o}))
119, 10mpanr2 435 . . . . 5 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1o ∪ {1o}))
1211ex 114 . . . 4 ((𝐴 ≈ 1o𝐵 ≈ {1o}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
136, 12sylan2 284 . . 3 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
14 df-2o 6361 . . . . 5 2o = suc 1o
15 df-suc 4331 . . . . 5 suc 1o = (1o ∪ {1o})
1614, 15eqtri 2178 . . . 4 2o = (1o ∪ {1o})
1716breq2i 3973 . . 3 ((𝐴𝐵) ≈ 2o ↔ (𝐴𝐵) ≈ (1o ∪ {1o}))
1813, 17syl6ibr 161 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2o))
19 en1 6741 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 6741 . . 3 (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21 1nen2 6803 . . . . . . . . . . . . 13 ¬ 1o ≈ 2o
2221a1i 9 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ¬ 1o ≈ 2o)
23 unidm 3250 . . . . . . . . . . . . . . . 16 ({𝑥} ∪ {𝑥}) = {𝑥}
24 sneq 3571 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2524uneq2d 3261 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2623, 25syl5reqr 2205 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27 vex 2715 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2827ensn1 6738 . . . . . . . . . . . . . . 15 {𝑥} ≈ 1o
2926, 28eqbrtrdi 4003 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1o)
3029ensymd 6725 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → 1o ≈ ({𝑥} ∪ {𝑦}))
31 entr 6726 . . . . . . . . . . . . 13 ((1o ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
3230, 31sylan 281 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
3322, 32mtand 655 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3433necon2ai 2381 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2o𝑥𝑦)
35 disjsn2 3622 . . . . . . . . . 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 3256 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3938breq1d 3975 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
40 ineq12 3303 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4140eqeq1d 2166 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4237, 39, 413imtr4d 202 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4342ex 114 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4443exlimdv 1799 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4544exlimiv 1578 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4645imp 123 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4719, 20, 46syl2anb 289 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4818, 47impbid 128 1 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128   ≠ wne 2327   ∪ cun 3100   ∩ cin 3101  ∅c0 3394  {csn 3560   class class class wbr 3965  Oncon0 4323  suc csuc 4325  1oc1o 6353  2oc2o 6354   ≈ cen 6680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1o 6360  df-2o 6361  df-er 6477  df-en 6683 This theorem is referenced by:  pr2nelem  7120  dju1p1e2  7126
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