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Theorem phplem2 8084
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
phplem2.1 𝐴 ∈ V
phplem2.2 𝐵 ∈ V
Assertion
Ref Expression
phplem2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem2
StepHypRef Expression
1 snex 4869 . . . . . 6 {⟨𝐵, 𝐴⟩} ∈ V
2 phplem2.2 . . . . . . 7 𝐵 ∈ V
3 phplem2.1 . . . . . . 7 𝐴 ∈ V
42, 3f1osn 6133 . . . . . 6 {⟨𝐵, 𝐴⟩}:{𝐵}–1-1-onto→{𝐴}
5 f1oen3g 7915 . . . . . 6 (({⟨𝐵, 𝐴⟩} ∈ V ∧ {⟨𝐵, 𝐴⟩}:{𝐵}–1-1-onto→{𝐴}) → {𝐵} ≈ {𝐴})
61, 4, 5mp2an 707 . . . . 5 {𝐵} ≈ {𝐴}
7 difss 3715 . . . . . . 7 (𝐴 ∖ {𝐵}) ⊆ 𝐴
83, 7ssexi 4763 . . . . . 6 (𝐴 ∖ {𝐵}) ∈ V
98enref 7932 . . . . 5 (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵})
106, 9pm3.2i 471 . . . 4 ({𝐵} ≈ {𝐴} ∧ (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵}))
11 incom 3783 . . . . . 6 ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ((𝐴 ∖ {𝐵}) ∩ {𝐴})
12 ssrin 3816 . . . . . . . . 9 ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ (𝐴 ∩ {𝐴}))
137, 12ax-mp 5 . . . . . . . 8 ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ (𝐴 ∩ {𝐴})
14 nnord 7020 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
15 orddisj 5721 . . . . . . . . 9 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
1614, 15syl 17 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
1713, 16syl5sseq 3632 . . . . . . 7 (𝐴 ∈ ω → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ ∅)
18 ss0 3946 . . . . . . 7 (((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ ∅ → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) = ∅)
1917, 18syl 17 . . . . . 6 (𝐴 ∈ ω → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) = ∅)
2011, 19syl5eq 2667 . . . . 5 (𝐴 ∈ ω → ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅)
21 disjdif 4012 . . . . 5 ({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅
2220, 21jctil 559 . . . 4 (𝐴 ∈ ω → (({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅ ∧ ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅))
23 unen 7984 . . . 4 ((({𝐵} ≈ {𝐴} ∧ (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵})) ∧ (({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅ ∧ ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅)) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
2410, 22, 23sylancr 694 . . 3 (𝐴 ∈ ω → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
2524adantr 481 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
26 uncom 3735 . . . 4 ({𝐵} ∪ (𝐴 ∖ {𝐵})) = ((𝐴 ∖ {𝐵}) ∪ {𝐵})
27 difsnid 4310 . . . 4 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
2826, 27syl5eq 2667 . . 3 (𝐵𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
2928adantl 482 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
30 phplem1 8083 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
3125, 29, 303brtr3d 4644 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148  cop 4154   class class class wbr 4613  Ord word 5681  suc csuc 5684  1-1-ontowf1o 5846  ωcom 7012  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-om 7013  df-en 7900
This theorem is referenced by:  phplem3  8085
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