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Theorem nolesgn2o 32161
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 1229 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → 𝐵 No )
2 nofv 32147 . . . . . 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 31931 . . . . 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 1247 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝑋 ∈ On)
7 fveq1 6331 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2777 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 466 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 471 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6349 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6349 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2813 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3115 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 466 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1128 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 756 . . . . . . . . . . . . 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 540 . . . . . . . . . . 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 540 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → ((𝐵𝑋) = 1𝑜 → ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜)))
2219, 21orim12d 945 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜))))
23223impia 1109 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜)))
24 3mix3 1416 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
25 3mix2 1415 . . . . . . . . . 10 (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
2624, 25jaoi 844 . . . . . . . . 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 6342 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6342 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 31977 . . . . . . . 8 ((𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋) ↔ (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
3127, 30sylibr 224 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋))
32 raleq 3287 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6332 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6332 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 4799 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥) ↔ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋)))
3632, 35anbi12d 616 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥)) ↔ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋))))
3736rspcev 3460 . . . . . . 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 1474 . . . . . 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 1246 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐵 No )
40 simp11 1245 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐴 No )
41 sltval 32137 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 573 . . . . . 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 247 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐵 <s 𝐴)
44433expia 1114 . . . 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 141 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → (¬ 𝐵 <s 𝐴 → (𝐵𝑋) = 2𝑜))
47463impia 1109 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 196  wa 382  wo 834  w3o 1070  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  c0 4063  {ctp 4320  cop 4322   class class class wbr 4786  cres 5251  Oncon0 5866  cfv 6031  1𝑜c1o 7706  2𝑜c2o 7707   No csur 32130   <s cslt 32131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7713  df-2o 7714  df-no 32133  df-slt 32134
This theorem is referenced by:  nolesgn2ores  32162
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