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Theorem phplem2 8689
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 5322 . . . . . 6 {⟨𝐵, 𝐴⟩} ∈ V
2 phplem2.2 . . . . . . 7 𝐵 ∈ V
3 phplem2.1 . . . . . . 7 𝐴 ∈ V
42, 3f1osn 6647 . . . . . 6 {⟨𝐵, 𝐴⟩}:{𝐵}–1-1-onto→{𝐴}
5 f1oen3g 8517 . . . . . 6 (({⟨𝐵, 𝐴⟩} ∈ V ∧ {⟨𝐵, 𝐴⟩}:{𝐵}–1-1-onto→{𝐴}) → {𝐵} ≈ {𝐴})
61, 4, 5mp2an 690 . . . . 5 {𝐵} ≈ {𝐴}
73difexi 5223 . . . . . 6 (𝐴 ∖ {𝐵}) ∈ V
87enref 8534 . . . . 5 (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵})
96, 8pm3.2i 473 . . . 4 ({𝐵} ≈ {𝐴} ∧ (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵}))
10 incom 4176 . . . . . 6 ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ((𝐴 ∖ {𝐵}) ∩ {𝐴})
11 difss 4106 . . . . . . . . 9 (𝐴 ∖ {𝐵}) ⊆ 𝐴
12 ssrin 4208 . . . . . . . . 9 ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ (𝐴 ∩ {𝐴}))
1311, 12ax-mp 5 . . . . . . . 8 ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ (𝐴 ∩ {𝐴})
14 nnord 7580 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
15 orddisj 6222 . . . . . . . . 9 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
1614, 15syl 17 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
1713, 16sseqtrid 4017 . . . . . . 7 (𝐴 ∈ ω → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ ∅)
18 ss0 4350 . . . . . . 7 (((𝐴 ∖ {𝐵}) ∩ {𝐴}) ⊆ ∅ → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) = ∅)
1917, 18syl 17 . . . . . 6 (𝐴 ∈ ω → ((𝐴 ∖ {𝐵}) ∩ {𝐴}) = ∅)
2010, 19syl5eq 2866 . . . . 5 (𝐴 ∈ ω → ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅)
21 disjdif 4419 . . . . 5 ({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅
2220, 21jctil 522 . . . 4 (𝐴 ∈ ω → (({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅ ∧ ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅))
23 unen 8588 . . . 4 ((({𝐵} ≈ {𝐴} ∧ (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {𝐵})) ∧ (({𝐵} ∩ (𝐴 ∖ {𝐵})) = ∅ ∧ ({𝐴} ∩ (𝐴 ∖ {𝐵})) = ∅)) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
249, 22, 23sylancr 589 . . 3 (𝐴 ∈ ω → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
2524adantr 483 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ≈ ({𝐴} ∪ (𝐴 ∖ {𝐵})))
26 uncom 4127 . . . 4 ({𝐵} ∪ (𝐴 ∖ {𝐵})) = ((𝐴 ∖ {𝐵}) ∪ {𝐵})
27 difsnid 4735 . . . 4 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
2826, 27syl5eq 2866 . . 3 (𝐵𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
2928adantl 484 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
30 phplem1 8688 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
3125, 29, 303brtr3d 5088 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  cdif 3931  cun 3932  cin 3933  wss 3934  c0 4289  {csn 4559  cop 4565   class class class wbr 5057  Ord word 6183  suc csuc 6186  1-1-ontowf1o 6347  ωcom 7572  cen 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-om 7573  df-en 8502
This theorem is referenced by:  phplem3  8690
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