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Theorem toponss 13879
Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
toponss ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)

Proof of Theorem toponss
StepHypRef Expression
1 elssuni 3849 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 277 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 13868 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 276 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtrrd 3206 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wss 3141   cuni 3821  cfv 5228  TopOnctopon 13863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-topon 13864
This theorem is referenced by:  iscnp3  14056  cnntr  14078  cncnp  14083  tx1cn  14122  tx2cn  14123  txcnp  14124  mopnss  14303  xmettx  14363
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