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Theorem oaf1o 8172
 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 8143 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
2 oaword1 8161 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑥))
3 ontri1 6193 . . . . . 6 ((𝐴 ∈ On ∧ (𝐴 +o 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
41, 3syldan 594 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
52, 4mpbid 235 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
61, 5eldifd 3892 . . 3 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
76ralrimiva 3149 . 2 (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
8 simpl 486 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9 eldifi 4054 . . . . . 6 (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
109adantl 485 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11 eldifn 4055 . . . . . . 7 (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦𝐴)
1211adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦𝐴)
13 ontri1 6193 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1410, 13syldan 594 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
1512, 14mpbird 260 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴𝑦)
16 oawordeu 8164 . . . . 5 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝑦) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
178, 10, 15, 16syl21anc 836 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
18 eqcom 2805 . . . . 5 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
1918reubii 3344 . . . 4 (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
2017, 19sylib 221 . . 3 ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
2120ralrimiva 3149 . 2 (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
22 eqid 2798 . . 3 (𝑥 ∈ On ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +o 𝑥))
2322f1ompt 6852 . 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 586 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃!wreu 3108   ∖ cdif 3878   ⊆ wss 3881   ↦ cmpt 5110  Oncon0 6159  –1-1-onto→wf1o 6323  (class class class)co 7135   +o coa 8082 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089 This theorem is referenced by:  oacomf1olem  8173
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