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Theorem harval3 41817
Description: (harβ€˜π΄) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
Assertion
Ref Expression
harval3 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ ran card ∣ 𝐴 β‰Ί π‘₯})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem harval3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 harval2 9934 . 2 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝐴 β‰Ί 𝑦})
2 vex 3450 . . . . . 6 π‘₯ ∈ V
32a1i 11 . . . . 5 (𝐴 ∈ dom card β†’ π‘₯ ∈ V)
4 elrncard 41816 . . . . . . . . 9 (π‘₯ ∈ ran card ↔ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ Β¬ 𝑦 β‰ˆ π‘₯))
54simplbi 499 . . . . . . . 8 (π‘₯ ∈ ran card β†’ π‘₯ ∈ On)
65anim1i 616 . . . . . . 7 ((π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯) β†’ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯))
7 eleq1 2826 . . . . . . . 8 (𝑦 = π‘₯ β†’ (𝑦 ∈ On ↔ π‘₯ ∈ On))
8 breq2 5110 . . . . . . . 8 (𝑦 = π‘₯ β†’ (𝐴 β‰Ί 𝑦 ↔ 𝐴 β‰Ί π‘₯))
97, 8anbi12d 632 . . . . . . 7 (𝑦 = π‘₯ β†’ ((𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦) ↔ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)))
106, 9syl5ibr 246 . . . . . 6 (𝑦 = π‘₯ β†’ ((π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯) β†’ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)))
1110adantl 483 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑦 = π‘₯) β†’ ((π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯) β†’ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)))
12 ssidd 3968 . . . . 5 (𝐴 ∈ dom card β†’ π‘₯ βŠ† π‘₯)
133, 11, 12intabssd 41798 . . . 4 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)} βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)})
14 vex 3450 . . . . . . 7 𝑦 ∈ V
1514inex1 5275 . . . . . 6 (𝑦 ∩ (cardβ€˜π‘¦)) ∈ V
1615a1i 11 . . . . 5 (𝐴 ∈ dom card β†’ (𝑦 ∩ (cardβ€˜π‘¦)) ∈ V)
17 oncardid 9893 . . . . . . . . . . . 12 (𝑦 ∈ On β†’ (cardβ€˜π‘¦) β‰ˆ 𝑦)
1817ensymd 8946 . . . . . . . . . . 11 (𝑦 ∈ On β†’ 𝑦 β‰ˆ (cardβ€˜π‘¦))
19 sdomentr 9056 . . . . . . . . . . . 12 ((𝐴 β‰Ί 𝑦 ∧ 𝑦 β‰ˆ (cardβ€˜π‘¦)) β†’ 𝐴 β‰Ί (cardβ€˜π‘¦))
2019a1i 11 . . . . . . . . . . 11 (𝑦 ∈ On β†’ ((𝐴 β‰Ί 𝑦 ∧ 𝑦 β‰ˆ (cardβ€˜π‘¦)) β†’ 𝐴 β‰Ί (cardβ€˜π‘¦)))
2118, 20mpan2d 693 . . . . . . . . . 10 (𝑦 ∈ On β†’ (𝐴 β‰Ί 𝑦 β†’ 𝐴 β‰Ί (cardβ€˜π‘¦)))
22 df-card 9876 . . . . . . . . . . . 12 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
2322funmpt2 6541 . . . . . . . . . . 11 Fun card
24 onenon 9886 . . . . . . . . . . 11 (𝑦 ∈ On β†’ 𝑦 ∈ dom card)
25 fvelrn 7028 . . . . . . . . . . 11 ((Fun card ∧ 𝑦 ∈ dom card) β†’ (cardβ€˜π‘¦) ∈ ran card)
2623, 24, 25sylancr 588 . . . . . . . . . 10 (𝑦 ∈ On β†’ (cardβ€˜π‘¦) ∈ ran card)
2721, 26jctild 527 . . . . . . . . 9 (𝑦 ∈ On β†’ (𝐴 β‰Ί 𝑦 β†’ ((cardβ€˜π‘¦) ∈ ran card ∧ 𝐴 β‰Ί (cardβ€˜π‘¦))))
2827adantl 483 . . . . . . . 8 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ (𝐴 β‰Ί 𝑦 β†’ ((cardβ€˜π‘¦) ∈ ran card ∧ 𝐴 β‰Ί (cardβ€˜π‘¦))))
29 simpl 484 . . . . . . . . . 10 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)))
30 cardonle 9894 . . . . . . . . . . . 12 (𝑦 ∈ On β†’ (cardβ€˜π‘¦) βŠ† 𝑦)
3130adantl 483 . . . . . . . . . . 11 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ (cardβ€˜π‘¦) βŠ† 𝑦)
32 sseqin2 4176 . . . . . . . . . . 11 ((cardβ€˜π‘¦) βŠ† 𝑦 ↔ (𝑦 ∩ (cardβ€˜π‘¦)) = (cardβ€˜π‘¦))
3331, 32sylib 217 . . . . . . . . . 10 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ (𝑦 ∩ (cardβ€˜π‘¦)) = (cardβ€˜π‘¦))
3429, 33eqtrd 2777 . . . . . . . . 9 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ π‘₯ = (cardβ€˜π‘¦))
35 eleq1 2826 . . . . . . . . . 10 (π‘₯ = (cardβ€˜π‘¦) β†’ (π‘₯ ∈ ran card ↔ (cardβ€˜π‘¦) ∈ ran card))
36 breq2 5110 . . . . . . . . . 10 (π‘₯ = (cardβ€˜π‘¦) β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί (cardβ€˜π‘¦)))
3735, 36anbi12d 632 . . . . . . . . 9 (π‘₯ = (cardβ€˜π‘¦) β†’ ((π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯) ↔ ((cardβ€˜π‘¦) ∈ ran card ∧ 𝐴 β‰Ί (cardβ€˜π‘¦))))
3834, 37syl 17 . . . . . . . 8 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ ((π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯) ↔ ((cardβ€˜π‘¦) ∈ ran card ∧ 𝐴 β‰Ί (cardβ€˜π‘¦))))
3928, 38sylibrd 259 . . . . . . 7 ((π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) ∧ 𝑦 ∈ On) β†’ (𝐴 β‰Ί 𝑦 β†’ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)))
4039expimpd 455 . . . . . 6 (π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦)) β†’ ((𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦) β†’ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)))
4140adantl 483 . . . . 5 ((𝐴 ∈ dom card ∧ π‘₯ = (𝑦 ∩ (cardβ€˜π‘¦))) β†’ ((𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦) β†’ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)))
42 inss1 4189 . . . . . 6 (𝑦 ∩ (cardβ€˜π‘¦)) βŠ† 𝑦
4342a1i 11 . . . . 5 (𝐴 ∈ dom card β†’ (𝑦 ∩ (cardβ€˜π‘¦)) βŠ† 𝑦)
4416, 41, 43intabssd 41798 . . . 4 (𝐴 ∈ dom card β†’ ∩ {π‘₯ ∣ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)} βŠ† ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)})
4513, 44eqssd 3962 . . 3 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)} = ∩ {π‘₯ ∣ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)})
46 df-rab 3409 . . . 4 {𝑦 ∈ On ∣ 𝐴 β‰Ί 𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)}
4746inteqi 4912 . . 3 ∩ {𝑦 ∈ On ∣ 𝐴 β‰Ί 𝑦} = ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦)}
48 df-rab 3409 . . . 4 {π‘₯ ∈ ran card ∣ 𝐴 β‰Ί π‘₯} = {π‘₯ ∣ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)}
4948inteqi 4912 . . 3 ∩ {π‘₯ ∈ ran card ∣ 𝐴 β‰Ί π‘₯} = ∩ {π‘₯ ∣ (π‘₯ ∈ ran card ∧ 𝐴 β‰Ί π‘₯)}
5045, 47, 493eqtr4g 2802 . 2 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝐴 β‰Ί 𝑦} = ∩ {π‘₯ ∈ ran card ∣ 𝐴 β‰Ί π‘₯})
511, 50eqtrd 2777 1 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ ran card ∣ 𝐴 β‰Ί π‘₯})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3408  Vcvv 3446   ∩ cin 3910   βŠ† wss 3911  βˆ© cint 4908   class class class wbr 5106  dom cdm 5634  ran crn 5635  Oncon0 6318  Fun wfun 6491  β€˜cfv 6497   β‰ˆ cen 8881   β‰Ί csdm 8883  harchar 9493  cardccrd 9872
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-oi 9447  df-har 9494  df-card 9876
This theorem is referenced by:  harval3on  41818
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