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Theorem oaf1o 7589
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 7561 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
2 oaword1 7578 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
3 ontri1 5719 . . . . . 6 ((𝐴 ∈ On ∧ (𝐴 +𝑜 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
41, 3syldan 487 . . . . 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 3571 . . 3 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴))
76ralrimiva 2965 . 2 (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴))
8 simpl 473 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9 eldifi 3715 . . . . . 6 (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
109adantl 482 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11 eldifn 3716 . . . . . . 7 (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦𝐴)
1211adantl 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦𝐴)
13 ontri1 5719 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1410, 13syldan 487 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1512, 14mpbird 247 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴𝑦)
16 oawordeu 7581 . . . . 5 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝑦) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦)
178, 10, 15, 16syl21anc 1322 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦)
18 eqcom 2633 . . . . 5 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
1918reubii 3122 . . . 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 2965 . 2 (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥))
22 eqid 2626 . . 3 (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥))
2322f1ompt 6339 . 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 697 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 384   = wceq 1480  wcel 1992  wral 2912  ∃!wreu 2914  cdif 3557  wss 3560  cmpt 4678  Oncon0 5685  1-1-ontowf1o 5849  (class class class)co 6605   +𝑜 coa 7503
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510
This theorem is referenced by:  oacomf1olem  7590
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