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Theorem omssrncard 43558
Description: All natural numbers are cardinals. (Contributed by RP, 1-Oct-2023.)
Assertion
Ref Expression
omssrncard ω ⊆ ran card

Proof of Theorem omssrncard
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnon 7894 . . 3 (𝑥 ∈ ω → 𝑥 ∈ On)
2 onelon 6408 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3 simpl 482 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑥 ∈ On)
4 simpr 484 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
5 onelpss 6423 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
65biimpa 476 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝑦𝑥𝑦𝑥))
72, 3, 4, 6syl21anc 837 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦𝑥𝑦𝑥))
8 df-pss 3970 . . . . . . . . . 10 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
97, 8sylibr 234 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
109ex 412 . . . . . . . 8 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
111, 10syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑦𝑥𝑦𝑥))
1211imdistani 568 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦𝑥) → (𝑥 ∈ ω ∧ 𝑦𝑥))
13 php 9248 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑥𝑦)
1412, 13syl 17 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑥𝑦)
15 ensymb 9043 . . . . 5 (𝑥𝑦𝑦𝑥)
1614, 15sylnib 328 . . . 4 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑦𝑥)
1716ralrimiva 3145 . . 3 (𝑥 ∈ ω → ∀𝑦𝑥 ¬ 𝑦𝑥)
18 elrncard 43555 . . 3 (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 ¬ 𝑦𝑥))
191, 17, 18sylanbrc 583 . 2 (𝑥 ∈ ω → 𝑥 ∈ ran card)
2019ssriv 3986 1 ω ⊆ ran card
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2107  wne 2939  wral 3060  wss 3950  wpss 3951   class class class wbr 5142  ran crn 5685  Oncon0 6383  ωcom 7888  cen 8983  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980
This theorem is referenced by:  0iscard  43559  1iscard  43560  nna1iscard  43563
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