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Theorem nolesgn2o 32787
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 1185 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → 𝐵 No )
2 nofv 32773 . . . . . 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 32550 . . . . 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 1198 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝑋 ∈ On)
7 fveq1 6537 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2801 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 481 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 485 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6558 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6558 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2839 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3149 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 481 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1127 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 769 . . . . . . . . . . . . 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 551 . . . . . . . . . . 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 551 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = 1o → ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
2219, 21orim12d 959 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o))))
23223impia 1110 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
24 3mix3 1325 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
25 3mix2 1324 . . . . . . . . . 10 (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2624, 25jaoi 852 . . . . . . . . 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 6551 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6551 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 32593 . . . . . . . 8 ((𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋) ↔ (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
3127, 30sylibr 235 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))
32 raleq 3365 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6538 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6538 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 4975 . . . . . . . . 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 3559 . . . . . . 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 833 . . . . . 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 1197 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐵 No )
40 simp11 1196 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐴 No )
41 sltval 32763 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 584 . . . . . 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 258 . . . . 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 147 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ 𝐵 <s 𝐴 → (𝐵𝑋) = 2o))
47463impia 1110 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 207  wa 396  wo 842  w3o 1079  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wrex 3106  c0 4211  {ctp 4476  cop 4478   class class class wbr 4962  cres 5445  Oncon0 6066  cfv 6225  1oc1o 7946  2oc2o 7947   No csur 32756   <s cslt 32757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-1o 7953  df-2o 7954  df-no 32759  df-slt 32760
This theorem is referenced by:  nolesgn2ores  32788
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