Theorem bday11on 28275

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 6836	. . . . 5 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
2	1	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
3		onleft 28270	. . . . 5 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
4	3	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
5		onleft 28270	. . . . 5 ⊢ (𝐵 ∈ Ons → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
6	5	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
7	2, 4, 6	3eqtr3d 2780	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( L ‘𝐴) = ( L ‘𝐵))
8	7	oveq1d 7377	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) |s ∅) = (( L ‘𝐵) |s ∅))
9		oncutleft 28273	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
10	9	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = (( L ‘𝐴) |s ∅))
11		oncutleft 28273	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
12	11	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 = (( L ‘𝐵) |s ∅))
13	8, 10, 12	3eqtr4d 2782	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ‘cfv 6494  (class class class)co 7362   bday cbday 27623   |s ccuts 27769   O cold 27833   L cleft 27835  On_scons 28261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-1o 8400  df-2o 8401  df-no 27624  df-lts 27625  df-bday 27626  df-slts 27768  df-cuts 27770  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-ons 28262

This theorem is referenced by:  oniso  28281  bdayn0sf1o  28380  bdayfinbndlem1  28477