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Theorem nogesgn1o 27733
Description: Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then 𝐵(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
Assertion
Ref Expression
nogesgn1o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)

Proof of Theorem nogesgn1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1191 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → 𝐵 No )
2 nofv 27717 . . . . . 6 (𝐵 No → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
31, 2syl 17 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
4 3orel2 1483 . . . . 5 (¬ (𝐵𝑋) = 1o → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
53, 4syl5com 31 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ (𝐵𝑋) = 1o → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
6 simp13 1204 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝑋 ∈ On)
7 fveq1 6906 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87adantr 480 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
9 simpr 484 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
109fvresd 6927 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
119fvresd 6927 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
128, 10, 113eqtr3d 2783 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐴𝑦) = (𝐵𝑦))
1312ralrimiva 3144 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
1413adantr 480 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
15143ad2ant2 1133 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
16 simp2r 1199 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋) = 1o)
17 simp3 1137 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o))
1816, 17jca 511 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
19 andi 1009 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
2018, 19sylib 218 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
21 3mix1 1329 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
22 3mix2 1330 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2321, 22jaoi 857 . . . . . . . . 9 ((((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2420, 23syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
25 fvex 6920 . . . . . . . . 9 (𝐴𝑋) ∈ V
26 fvex 6920 . . . . . . . . 9 (𝐵𝑋) ∈ V
2725, 26brtp 5533 . . . . . . . 8 ((𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2824, 27sylibr 234 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))
29 raleq 3321 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦)))
30 fveq2 6907 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
31 fveq2 6907 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
3230, 31breq12d 5161 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
3329, 32anbi12d 632 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))))
3433rspcev 3622 . . . . . . 7 ((𝑋 ∈ On ∧ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
356, 15, 28, 34syl12anc 837 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
36 simp11 1202 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 No )
37 simp12 1203 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐵 No )
38 sltval 27707 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
3936, 37, 38syl2anc 584 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
4035, 39mpbird 257 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 <s 𝐵)
41403expia 1120 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o) → 𝐴 <s 𝐵))
425, 41syld 47 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ (𝐵𝑋) = 1o𝐴 <s 𝐵))
4342con1d 145 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ 𝐴 <s 𝐵 → (𝐵𝑋) = 1o))
44433impia 1116 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  c0 4339  {ctp 4635  cop 4637   class class class wbr 5148  cres 5691  Oncon0 6386  cfv 6563  1oc1o 8498  2oc2o 8499   No csur 27699   <s cslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703
This theorem is referenced by:  nogesgn1ores  27734
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