Theorem bday11on 28678

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 6905	. . . . 5 ⊢ ((bday‘𝐴) = (bday‘𝐵) → (O‘(bday‘𝐴)) = (O‘(bday‘𝐵)))
2	1	3ad2ant3 1153	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → (O‘(bday‘𝐴)) = (O‘(bday‘𝐵)))
3		onleft 28673	. . . . 5 ⊢ (𝐴 ∈ Ons → (O‘(bday‘𝐴)) = (L‘𝐴))
4	3	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → (O‘(bday‘𝐴)) = (L‘𝐴))
5		onleft 28673	. . . . 5 ⊢ (𝐵 ∈ Ons → (O‘(bday‘𝐵)) = (L‘𝐵))
6	5	3ad2ant2 1152	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → (O‘(bday‘𝐵)) = (L‘𝐵))
7	2, 4, 6	3eqtr3d 2806	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → (L‘𝐴) = (L‘𝐵))
8	7	oveq1d 7455	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → ((L‘𝐴) |s ∅) = ((L‘𝐵) |s ∅))
9		oncutleft 28676	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 = ((L‘𝐴) |s ∅))
10	9	3ad2ant1 1151	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐴 = ((L‘𝐴) |s ∅))
11		oncutleft 28676	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 = ((L‘𝐵) |s ∅))
12	11	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐵 = ((L‘𝐵) |s ∅))
13	8, 10, 12	3eqtr4d 2808	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ‘cfv 6555  (class class class)co 7440  bdaycbday 28026   |s ccuts 28172  Ocold 28236  Lcleft 28238  On_scons 28664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-ons 28665

This theorem is used by:  oniso  28684  bdayn0sf1o  28783  bdayfinbndlem1  28880