Theorem bday11on 28279

Description: The birthday function is one-to-one over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)

Assertion

Ref	Expression
bday11on	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)

Proof of Theorem bday11on

Step	Hyp	Ref	Expression
1		fveq2 6831	. . . . 5 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
2	1	3ad2ant3 1142	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
3		onleft 28274	. . . . 5 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
4	3	3ad2ant1 1140	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
5		onleft 28274	. . . . 5 ⊢ (𝐵 ∈ Ons → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
6	5	3ad2ant2 1141	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
7	2, 4, 6	3eqtr3d 2784	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( L ‘𝐴) = ( L ‘𝐵))
8	7	oveq1d 7375	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) |s ∅) = (( L ‘𝐵) |s ∅))
9		oncutleft 28277	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
10	9	3ad2ant1 1140	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = (( L ‘𝐴) |s ∅))
11		oncutleft 28277	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
12	11	3ad2ant2 1141	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 = (( L ‘𝐵) |s ∅))
13	8, 10, 12	3eqtr4d 2786	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∅c0 4264  ‘cfv 6489  (class class class)co 7360   bday cbday 27627   |s ccuts 27773   O cold 27837   L cleft 27839  On_scons 28265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27628  df-lts 27629  df-bday 27630  df-slts 27772  df-cuts 27774  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-ons 28266

This theorem is referenced by:  oniso  28285  bdayn0sf1o  28384  bdayfinbndlem1  28481