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Theorem dif1ennnALT 9228
Description: Alternate proof of dif1ennn 9112 using ax-pow 5325. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dif1ennnALT ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1ennnALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 peano2 7832 . . . . 5 (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2 breq2 5114 . . . . . . 7 (𝑥 = suc 𝑀 → (𝐴𝑥𝐴 ≈ suc 𝑀))
32rspcev 3584 . . . . . 6 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 8923 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
53, 4sylibr 233 . . . . 5 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
61, 5sylan 581 . . . 4 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7 diffi 9130 . . . . 5 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8 isfi 8923 . . . . 5 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
97, 8sylib 217 . . . 4 (𝐴 ∈ Fin → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
106, 9syl 17 . . 3 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11103adant3 1133 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12 en2sn 8992 . . . . . . . 8 ((𝑋𝐴𝑥 ∈ V) → {𝑋} ≈ {𝑥})
1312elvd 3455 . . . . . . 7 (𝑋𝐴 → {𝑋} ≈ {𝑥})
14 nnord 7815 . . . . . . . 8 (𝑥 ∈ ω → Ord 𝑥)
15 orddisj 6360 . . . . . . . 8 (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
1614, 15syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17 incom 4166 . . . . . . . . . 10 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18 disjdif 4436 . . . . . . . . . 10 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
1917, 18eqtri 2765 . . . . . . . . 9 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20 unen 8997 . . . . . . . . . 10 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2120an4s 659 . . . . . . . . 9 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2219, 21mpanl2 700 . . . . . . . 8 (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2322expcom 415 . . . . . . 7 (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
2413, 16, 23syl2an 597 . . . . . 6 ((𝑋𝐴𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25243ad2antl3 1188 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26 difsnid 4775 . . . . . . . . 9 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27 df-suc 6328 . . . . . . . . . . 11 suc 𝑥 = (𝑥 ∪ {𝑥})
2827eqcomi 2746 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = suc 𝑥
2928a1i 11 . . . . . . . . 9 (𝑋𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
3026, 29breq12d 5123 . . . . . . . 8 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31303ad2ant3 1136 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
3231adantr 482 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33 ensym 8950 . . . . . . . . . . 11 (𝐴 ≈ suc 𝑀 → suc 𝑀𝐴)
34 entr 8953 . . . . . . . . . . . . 13 ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35 peano2 7832 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36 nneneq 9160 . . . . . . . . . . . . . 14 ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3735, 36sylan2 594 . . . . . . . . . . . . 13 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3834, 37imbitrid 243 . . . . . . . . . . . 12 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
3938expd 417 . . . . . . . . . . 11 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀𝐴 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4033, 39syl5 34 . . . . . . . . . 10 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
411, 40sylan 581 . . . . . . . . 9 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4241imp 408 . . . . . . . 8 (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4342an32s 651 . . . . . . 7 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44433adantl3 1169 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4532, 44sylbid 239 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
46 peano4 7834 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4746biimpd 228 . . . . . 6 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
48473ad2antl1 1186 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4925, 45, 483syld 60 . . . 4 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥𝑀 = 𝑥))
50 breq2 5114 . . . . 5 (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
5150biimprcd 250 . . . 4 ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝑀 = 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5249, 51sylcom 30 . . 3 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5352rexlimdva 3153 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5411, 53mpd 15 1 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3074  Vcvv 3448  cdif 3912  cun 3913  cin 3914  c0 4287  {csn 4591   class class class wbr 5110  Ord word 6321  suc csuc 6324  ωcom 7807  cen 8887  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-er 8655  df-en 8891  df-fin 8894
This theorem is referenced by: (None)
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