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Theorem oaf1o 8563
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 β†’ (π‘₯ ∈ On ↦ (𝐴 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐴))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem oaf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oacl 8535 . . . 4 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ (𝐴 +o π‘₯) ∈ On)
2 oaword1 8552 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ 𝐴 βŠ† (𝐴 +o π‘₯))
3 ontri1 6399 . . . . . 6 ((𝐴 ∈ On ∧ (𝐴 +o π‘₯) ∈ On) β†’ (𝐴 βŠ† (𝐴 +o π‘₯) ↔ Β¬ (𝐴 +o π‘₯) ∈ 𝐴))
41, 3syldan 592 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ (𝐴 βŠ† (𝐴 +o π‘₯) ↔ Β¬ (𝐴 +o π‘₯) ∈ 𝐴))
52, 4mpbid 231 . . . 4 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ Β¬ (𝐴 +o π‘₯) ∈ 𝐴)
61, 5eldifd 3960 . . 3 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ (𝐴 +o π‘₯) ∈ (On βˆ– 𝐴))
76ralrimiva 3147 . 2 (𝐴 ∈ On β†’ βˆ€π‘₯ ∈ On (𝐴 +o π‘₯) ∈ (On βˆ– 𝐴))
8 simpl 484 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ 𝐴 ∈ On)
9 eldifi 4127 . . . . . 6 (𝑦 ∈ (On βˆ– 𝐴) β†’ 𝑦 ∈ On)
109adantl 483 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ 𝑦 ∈ On)
11 eldifn 4128 . . . . . . 7 (𝑦 ∈ (On βˆ– 𝐴) β†’ Β¬ 𝑦 ∈ 𝐴)
1211adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ Β¬ 𝑦 ∈ 𝐴)
13 ontri1 6399 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) β†’ (𝐴 βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ 𝐴))
1410, 13syldan 592 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ (𝐴 βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ 𝐴))
1512, 14mpbird 257 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ 𝐴 βŠ† 𝑦)
16 oawordeu 8555 . . . . 5 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 βŠ† 𝑦) β†’ βˆƒ!π‘₯ ∈ On (𝐴 +o π‘₯) = 𝑦)
178, 10, 15, 16syl21anc 837 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ βˆƒ!π‘₯ ∈ On (𝐴 +o π‘₯) = 𝑦)
18 eqcom 2740 . . . . 5 ((𝐴 +o π‘₯) = 𝑦 ↔ 𝑦 = (𝐴 +o π‘₯))
1918reubii 3386 . . . 4 (βˆƒ!π‘₯ ∈ On (𝐴 +o π‘₯) = 𝑦 ↔ βˆƒ!π‘₯ ∈ On 𝑦 = (𝐴 +o π‘₯))
2017, 19sylib 217 . . 3 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ βˆƒ!π‘₯ ∈ On 𝑦 = (𝐴 +o π‘₯))
2120ralrimiva 3147 . 2 (𝐴 ∈ On β†’ βˆ€π‘¦ ∈ (On βˆ– 𝐴)βˆƒ!π‘₯ ∈ On 𝑦 = (𝐴 +o π‘₯))
22 eqid 2733 . . 3 (π‘₯ ∈ On ↦ (𝐴 +o π‘₯)) = (π‘₯ ∈ On ↦ (𝐴 +o π‘₯))
2322f1ompt 7111 . 2 ((π‘₯ ∈ On ↦ (𝐴 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐴) ↔ (βˆ€π‘₯ ∈ On (𝐴 +o π‘₯) ∈ (On βˆ– 𝐴) ∧ βˆ€π‘¦ ∈ (On βˆ– 𝐴)βˆƒ!π‘₯ ∈ On 𝑦 = (𝐴 +o π‘₯)))
247, 21, 23sylanbrc 584 1 (𝐴 ∈ On β†’ (π‘₯ ∈ On ↦ (𝐴 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒ!wreu 3375   βˆ– cdif 3946   βŠ† wss 3949   ↦ cmpt 5232  Oncon0 6365  β€“1-1-ontoβ†’wf1o 6543  (class class class)co 7409   +o coa 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-oadd 8470
This theorem is referenced by:  oacomf1olem  8564
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