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Theorem oaf1o 8514
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 8485 . . . 4 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ (𝐴 +o π‘₯) ∈ On)
2 oaword1 8503 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ 𝐴 βŠ† (𝐴 +o π‘₯))
3 ontri1 6355 . . . . . 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 3925 . . 3 ((𝐴 ∈ On ∧ π‘₯ ∈ On) β†’ (𝐴 +o π‘₯) ∈ (On βˆ– 𝐴))
76ralrimiva 3140 . 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 4090 . . . . . 6 (𝑦 ∈ (On βˆ– 𝐴) β†’ 𝑦 ∈ On)
109adantl 483 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ 𝑦 ∈ On)
11 eldifn 4091 . . . . . . 7 (𝑦 ∈ (On βˆ– 𝐴) β†’ Β¬ 𝑦 ∈ 𝐴)
1211adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ (On βˆ– 𝐴)) β†’ Β¬ 𝑦 ∈ 𝐴)
13 ontri1 6355 . . . . . . 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 8506 . . . . 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 3361 . . . 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 3140 . 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 7063 . 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 3061  βˆƒ!wreu 3350   βˆ– cdif 3911   βŠ† wss 3914   ↦ cmpt 5192  Oncon0 6321  β€“1-1-ontoβ†’wf1o 6499  (class class class)co 7361   +o coa 8413
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420
This theorem is referenced by:  oacomf1olem  8515
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