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Theorem cardne 9994
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Proof of Theorem cardne
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6937 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐡 ∈ dom card)
2 cardon 9973 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
32oneli 6486 . . . . . . . . 9 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 ∈ On)
4 breq1 5153 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐡 ↔ 𝐴 β‰ˆ 𝐡))
54onintss 6423 . . . . . . . . 9 (𝐴 ∈ On β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
76adantl 480 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
8 cardval3 9981 . . . . . . . . 9 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡})
98sseq1d 4011 . . . . . . . 8 (𝐡 ∈ dom card β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
109adantr 479 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
117, 10sylibrd 258 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΅) βŠ† 𝐴))
12 ontri1 6406 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
132, 3, 12sylancr 585 . . . . . . 7 (𝐴 ∈ (cardβ€˜π΅) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1413adantl 480 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1511, 14sylibd 238 . . . . 5 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1615con2d 134 . . . 4 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
1716ex 411 . . 3 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)))
1817pm2.43d 53 . 2 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
191, 18mpcom 38 1 (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2098  {crab 3428   βŠ† wss 3947  βˆ© cint 4951   class class class wbr 5150  dom cdm 5680  Oncon0 6372  β€˜cfv 6551   β‰ˆ cen 8965  cardccrd 9964
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-en 8969  df-card 9968
This theorem is referenced by:  carden2b  9996  cardlim  10001  cardsdomelir  10002
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