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Theorem nolesgn2o 27622
Description: Given 𝐴 less-than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nolesgn2o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)

Proof of Theorem nolesgn2o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1189 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → 𝐵 No )
2 nofv 27608 . . . . . 6 (𝐵 No → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
31, 2syl 17 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
4 3orel3 1481 . . . . 5 (¬ (𝐵𝑋) = 2o → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)))
53, 4syl5com 31 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ (𝐵𝑋) = 2o → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)))
6 simp13 1202 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝑋 ∈ On)
7 fveq1 6891 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2731 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 479 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 483 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6912 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6912 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2773 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3136 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 479 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1131 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 771 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (𝐴𝑋) = 2o)
1817a1d 25 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ → (𝐴𝑋) = 2o))
1918ancld 549 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ → ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2017a1d 25 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = 1o → (𝐴𝑋) = 2o))
2120ancld 549 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = 1o → ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
2219, 21orim12d 962 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o))))
23223impia 1114 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
24 3mix3 1329 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
25 3mix2 1328 . . . . . . . . . 10 (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2624, 25jaoi 855 . . . . . . . . 9 ((((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2723, 26syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
28 fvex 6905 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6905 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 5519 . . . . . . . 8 ((𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋) ↔ (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
3127, 30sylibr 233 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))
32 raleq 3312 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6892 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6892 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 5156 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥) ↔ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋)))
3632, 35anbi12d 630 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)) ↔ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))))
3736rspcev 3601 . . . . . . 7 ((𝑋 ∈ On ∧ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)))
386, 16, 31, 37syl12anc 835 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)))
39 simp12 1201 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐵 No )
40 simp11 1200 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐴 No )
41 sltval 27598 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 582 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥))))
4338, 42mpbird 256 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐵 <s 𝐴)
44433expia 1118 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o) → 𝐵 <s 𝐴))
455, 44syld 47 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ (𝐵𝑋) = 2o𝐵 <s 𝐴))
4645con1d 145 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ 𝐵 <s 𝐴 → (𝐵𝑋) = 2o))
47463impia 1114 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  wral 3051  wrex 3060  c0 4318  {ctp 4628  cop 4630   class class class wbr 5143  cres 5674  Oncon0 6364  cfv 6543  1oc1o 8478  2oc2o 8479   No csur 27591   <s cslt 27592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8485  df-2o 8486  df-no 27594  df-slt 27595
This theorem is referenced by:  nolesgn2ores  27623
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