Theorem bday11on 28206

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 6840	. . . . 5 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
2	1	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( O ‘( bday ‘𝐵)))
3		onsleft 28201	. . . . 5 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
4	3	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
5		onsleft 28201	. . . . 5 ⊢ (𝐵 ∈ Ons → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
6	5	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐵))
7	2, 4, 6	3eqtr3d 2772	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( L ‘𝐴) = ( L ‘𝐵))
8	7	oveq1d 7384	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) |s ∅) = (( L ‘𝐵) |s ∅))
9		onscutleft 28204	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
10	9	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = (( L ‘𝐴) |s ∅))
11		onscutleft 28204	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
12	11	3ad2ant2 1134	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 = (( L ‘𝐵) |s ∅))
13	8, 10, 12	3eqtr4d 2774	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4292  ‘cfv 6499  (class class class)co 7369   bday cbday 27586   |s cscut 27728   O cold 27788   L cleft 27790  On_scons 28192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-bday 27589  df-sslt 27727  df-scut 27729  df-made 27792  df-old 27793  df-left 27795  df-right 27796  df-ons 28193

This theorem is referenced by:  onsiso  28209  bdayn0sf1o  28299