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Mirrors > Home > ILE Home > Th. List > enpr2d | GIF version |
Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
enpr2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
enpr2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
enpr2d.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
enpr2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2d.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | ensn1g 6742 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
4 | enpr2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 1on 6370 | . . . . 5 ⊢ 1o ∈ On | |
6 | en2sn 6758 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o}) | |
7 | 4, 5, 6 | sylancl 410 | . . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o}) |
8 | enpr2d.3 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
9 | 8 | neqned 2334 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | disjsn2 3622 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
12 | 5 | onirri 4502 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
13 | 12 | a1i 9 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
14 | disjsn 3621 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
15 | 13, 14 | sylibr 133 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
16 | unen 6761 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
17 | 3, 7, 11, 15, 16 | syl22anc 1221 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) |
18 | df-pr 3567 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
19 | df-suc 4331 | . . 3 ⊢ suc 1o = (1o ∪ {1o}) | |
20 | 17, 18, 19 | 3brtr4g 3998 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
21 | df-2o 6364 | . 2 ⊢ 2o = suc 1o | |
22 | 20, 21 | breqtrrdi 4006 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 {cpr 3561 class class class wbr 3965 Oncon0 4323 suc csuc 4325 1oc1o 6356 2oc2o 6357 ≈ cen 6683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-1o 6363 df-2o 6364 df-er 6480 df-en 6686 |
This theorem is referenced by: (None) |
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