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Theorem oaf1o 7688
Description: Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
oaf1o (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem oaf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oacl 7660 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
2 oaword1 7677 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
3 ontri1 5795 . . . . . 6 ((𝐴 ∈ On ∧ (𝐴 +𝑜 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
41, 3syldan 486 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
52, 4mpbid 222 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
61, 5eldifd 3618 . . 3 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴))
76ralrimiva 2995 . 2 (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴))
8 simpl 472 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9 eldifi 3765 . . . . . 6 (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
109adantl 481 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11 eldifn 3766 . . . . . . 7 (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦𝐴)
1211adantl 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦𝐴)
13 ontri1 5795 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1410, 13syldan 486 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1512, 14mpbird 247 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴𝑦)
16 oawordeu 7680 . . . . 5 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝑦) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦)
178, 10, 15, 16syl21anc 1365 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦)
18 eqcom 2658 . . . . 5 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
1918reubii 3158 . . . 4 (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥))
2017, 19sylib 208 . . 3 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥))
2120ralrimiva 2995 . 2 (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥))
22 eqid 2651 . . 3 (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥))
2322f1ompt 6422 . 2 ((𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴) ↔ (∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴) ∧ ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)))
247, 21, 23sylanbrc 699 1 (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ∃!wreu 2943  cdif 3604  wss 3607  cmpt 4762  Oncon0 5761  1-1-ontowf1o 5925  (class class class)co 6690   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  oacomf1olem  7689
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