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Theorem infpssrlem5 9444
Description: Lemma for infpssr 9445. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Hypotheses
Ref Expression
infpssrlem.a (𝜑𝐵𝐴)
infpssrlem.c (𝜑𝐹:𝐵1-1-onto𝐴)
infpssrlem.d (𝜑𝐶 ∈ (𝐴𝐵))
infpssrlem.e 𝐺 = (rec(𝐹, 𝐶) ↾ ω)
Assertion
Ref Expression
infpssrlem5 (𝜑 → (𝐴𝑉 → ω ≼ 𝐴))

Proof of Theorem infpssrlem5
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infpssrlem.a . . . 4 (𝜑𝐵𝐴)
2 infpssrlem.c . . . 4 (𝜑𝐹:𝐵1-1-onto𝐴)
3 infpssrlem.d . . . 4 (𝜑𝐶 ∈ (𝐴𝐵))
4 infpssrlem.e . . . 4 𝐺 = (rec(𝐹, 𝐶) ↾ ω)
51, 2, 3, 4infpssrlem3 9442 . . 3 (𝜑𝐺:ω⟶𝐴)
6 simpll 783 . . . . . . . . . 10 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑏𝑐) → 𝜑)
7 simplrr 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑏𝑐) → 𝑐 ∈ ω)
8 simpr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑏𝑐) → 𝑏𝑐)
91, 2, 3, 4infpssrlem4 9443 . . . . . . . . . 10 ((𝜑𝑐 ∈ ω ∧ 𝑏𝑐) → (𝐺𝑐) ≠ (𝐺𝑏))
106, 7, 8, 9syl3anc 1494 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑏𝑐) → (𝐺𝑐) ≠ (𝐺𝑏))
1110necomd 3054 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑏𝑐) → (𝐺𝑏) ≠ (𝐺𝑐))
12 simpll 783 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑐𝑏) → 𝜑)
13 simplrl 795 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑐𝑏) → 𝑏 ∈ ω)
14 simpr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑐𝑏) → 𝑐𝑏)
151, 2, 3, 4infpssrlem4 9443 . . . . . . . . 9 ((𝜑𝑏 ∈ ω ∧ 𝑐𝑏) → (𝐺𝑏) ≠ (𝐺𝑐))
1612, 13, 14, 15syl3anc 1494 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑐𝑏) → (𝐺𝑏) ≠ (𝐺𝑐))
1711, 16jaodan 985 . . . . . . 7 (((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ (𝑏𝑐𝑐𝑏)) → (𝐺𝑏) ≠ (𝐺𝑐))
1817ex 403 . . . . . 6 ((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ((𝑏𝑐𝑐𝑏) → (𝐺𝑏) ≠ (𝐺𝑐)))
1918necon2bd 3015 . . . . 5 ((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ((𝐺𝑏) = (𝐺𝑐) → ¬ (𝑏𝑐𝑐𝑏)))
20 nnord 7334 . . . . . . 7 (𝑏 ∈ ω → Ord 𝑏)
21 nnord 7334 . . . . . . 7 (𝑐 ∈ ω → Ord 𝑐)
22 ordtri3 5999 . . . . . . 7 ((Ord 𝑏 ∧ Ord 𝑐) → (𝑏 = 𝑐 ↔ ¬ (𝑏𝑐𝑐𝑏)))
2320, 21, 22syl2an 589 . . . . . 6 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 = 𝑐 ↔ ¬ (𝑏𝑐𝑐𝑏)))
2423adantl 475 . . . . 5 ((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → (𝑏 = 𝑐 ↔ ¬ (𝑏𝑐𝑐𝑏)))
2519, 24sylibrd 251 . . . 4 ((𝜑 ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ((𝐺𝑏) = (𝐺𝑐) → 𝑏 = 𝑐))
2625ralrimivva 3180 . . 3 (𝜑 → ∀𝑏 ∈ ω ∀𝑐 ∈ ω ((𝐺𝑏) = (𝐺𝑐) → 𝑏 = 𝑐))
27 dff13 6767 . . 3 (𝐺:ω–1-1𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑏 ∈ ω ∀𝑐 ∈ ω ((𝐺𝑏) = (𝐺𝑐) → 𝑏 = 𝑐)))
285, 26, 27sylanbrc 578 . 2 (𝜑𝐺:ω–1-1𝐴)
29 f1domg 8242 . 2 (𝐴𝑉 → (𝐺:ω–1-1𝐴 → ω ≼ 𝐴))
3028, 29syl5com 31 1 (𝜑 → (𝐴𝑉 → ω ≼ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  wne 2999  wral 3117  cdif 3795  wss 3798   class class class wbr 4873  ccnv 5341  cres 5344  Ord word 5962  wf 6119  1-1wf1 6120  1-1-ontowf1o 6122  cfv 6123  ωcom 7326  reccrdg 7771  cdom 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-dom 8224
This theorem is referenced by:  infpssr  9445
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