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Theorem harval2 10028
Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harval2 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem harval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 harval 9591 . . . . . . 7 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
21adantr 479 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
3 sdomel 9155 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ π‘₯ ∈ On) β†’ (𝑦 β‰Ί π‘₯ β†’ 𝑦 ∈ π‘₯))
4 domsdomtr 9143 . . . . . . . . . . . 12 ((𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯) β†’ 𝑦 β‰Ί π‘₯)
53, 4impel 504 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ π‘₯ ∈ On) ∧ (𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
65an4s 658 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
76ancoms 457 . . . . . . . . 9 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴)) β†’ 𝑦 ∈ π‘₯)
873impb 1112 . . . . . . . 8 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ 𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) β†’ 𝑦 ∈ π‘₯)
98rabssdv 4072 . . . . . . 7 ((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
109adantl 480 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
112, 10eqsstrd 4020 . . . . 5 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) βŠ† π‘₯)
1211expr 455 . . . 4 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
1312ralrimiva 3143 . . 3 (𝐴 ∈ dom card β†’ βˆ€π‘₯ ∈ On (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
14 ssintrab 4978 . . 3 ((harβ€˜π΄) βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ↔ βˆ€π‘₯ ∈ On (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
1513, 14sylibr 233 . 2 (𝐴 ∈ dom card β†’ (harβ€˜π΄) βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
16 breq2 5156 . . . 4 (π‘₯ = (harβ€˜π΄) β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί (harβ€˜π΄)))
17 harcl 9590 . . . . 5 (harβ€˜π΄) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ On)
19 harsdom 10026 . . . 4 (𝐴 ∈ dom card β†’ 𝐴 β‰Ί (harβ€˜π΄))
2016, 18, 19elrabd 3686 . . 3 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
21 intss1 4970 . . 3 ((harβ€˜π΄) ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† (harβ€˜π΄))
2220, 21syl 17 . 2 (𝐴 ∈ dom card β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† (harβ€˜π΄))
2315, 22eqssd 3999 1 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   βŠ† wss 3949  βˆ© cint 4953   class class class wbr 5152  dom cdm 5682  Oncon0 6374  β€˜cfv 6553   β‰Ό cdom 8968   β‰Ί csdm 8969  harchar 9587  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-oi 9541  df-har 9588  df-card 9970
This theorem is referenced by:  harsucnn  10029  alephnbtwn  10102  harval3  42999  aleph1min  43018
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