Theorem bday11on 28469

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 6881	. . . . 5 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
2	1	3ad2ant3 1152	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
3		onleft 28464	. . . . 5 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
4	3	3ad2ant1 1150	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
5		onleft 28464	. . . . 5 ⊢ (𝐵 ∈ Ons → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
6	5	3ad2ant2 1151	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
7	2, 4, 6	3eqtr3d 2805	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( L ‘𝐴) = ( L ‘𝐵))
8	7	oveq1d 7427	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) |s ∅) = (( L ‘𝐵) |s ∅))
9		oncutleft 28467	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
10	9	3ad2ant1 1150	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = (( L ‘𝐴) |s ∅))
11		oncutleft 28467	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
12	11	3ad2ant2 1151	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 = (( L ‘𝐵) |s ∅))
13	8, 10, 12	3eqtr4d 2807	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∅c0 4285  ‘cfv 6536  (class class class)co 7412   bday cbday 27817   |s ccuts 27963   O cold 28027   L cleft 28029  On_scons 28455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-1o 8451  df-2o 8452  df-no 27818  df-lts 27819  df-bday 27820  df-slts 27962  df-cuts 27964  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-ons 28456

This theorem is used by:  oniso  28475  bdayn0sf1o  28574  bdayfinbndlem1  28671