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Theorem nolesgn2o 32152
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 1230 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → 𝐵 No )
2 nofv 32138 . . . . . 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 31922 . . . . 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 1248 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝑋 ∈ On)
7 fveq1 6353 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2767 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 472 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 479 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6371 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6371 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2803 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3105 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 472 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1129 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 813 . . . . . . . . . . . . 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 577 . . . . . . . . . . 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 577 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → ((𝐵𝑋) = 1𝑜 → ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜)))
2219, 21orim12d 919 . . . . . . . . . 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 1417 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
25 3mix2 1416 . . . . . . . . . 10 (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
2624, 25jaoi 393 . . . . . . . . 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 6364 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6364 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 31968 . . . . . . . 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 3278 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6354 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6354 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 4818 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥) ↔ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋)))
3632, 35anbi12d 749 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥)) ↔ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋))))
3736rspcev 3450 . . . . . . 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 1475 . . . . . 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 1247 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐵 No )
40 simp11 1246 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐴 No )
41 sltval 32128 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 696 . . . . . 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 1115 . . . 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 139 . 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 196  wo 382  wa 383  w3o 1071  w3a 1072   = wceq 1632  wcel 2140  wral 3051  wrex 3052  c0 4059  {ctp 4326  cop 4328   class class class wbr 4805  cres 5269  Oncon0 5885  cfv 6050  1𝑜c1o 7724  2𝑜c2o 7725   No csur 32121   <s cslt 32122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ord 5888  df-on 5889  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-1o 7731  df-2o 7732  df-no 32124  df-slt 32125
This theorem is referenced by:  nolesgn2ores  32153
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